arXiv:1310.8279v4 [math.CT] 13 Oct 2015 · Adjunction data. In [25], we develop the theory of...

HOMOTOPY COHERENT ADJUNCTIONS AND THE FORMALTHEORY OF MONADS

EMILY RIEHL AND DOMINIC VERITY

Abstract. In this paper, we introduce a cofibrant simplicial category that we call thefree homotopy coherent adjunction and characterise its n-arrows using a graphical calculusthat we develop here. The hom-spaces are appropriately fibrant, indeed are nerves ofcategories, which indicates that all of the expected coherence equations in each dimensionare present. To justify our terminology, we prove that any adjunction of quasi-categoriesextends to a homotopy coherent adjunction and furthermore that these extensions arehomotopically unique in the sense that the relevant spaces of extensions are contractibleKan complexes.

We extract several simplicial functors from the free homotopy coherent adjunction andshow that quasi-categories are closed under weighted limits with these weights. Theseweighted limits are used to define the homotopy coherent monadic adjunction associatedto a homotopy coherent monad. We show that each vertex in the quasi-category of al-gebras for a homotopy coherent monad is a codescent object of a canonical diagram offree algebras. To conclude, we prove the quasi-categorical monadicity theorem, describingconditions under which the canonical comparison functor from a homotopy coherent ad-junction to the associated monadic adjunction is an equivalence of quasi-categories. Ourproofs reveal that a mild variant of Beck’s argument is “all in the weights”—much of itindependent of the quasi-categorical context.

Contents

1. Introduction 21.1. Adjunction data 21.2. The free homotopy coherent adjunction 41.3. Weighted limits and the formal theory of monads 51.4. Acknowledgments 72. Simplicial computads 72.1. Simplicial categories and simplicial computads 72.2. Simplicial subcomputads 103. The generic adjunction 123.1. A graphical calculus for the simplicial category Adj 123.2. The simplicial category Adj as a 2-category 213.3. The 2-categorical universal property of Adj 23

Date: October 14, 2015.2010 Mathematics Subject Classification. Primary 18G55, 55U35, 55U40; Secondary 18A05, 18D20,

18G30, 55U10.

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2 RIEHL AND VERITY

4. Adjunction data 284.1. Fillable arrows 284.2. Parental subcomputads 304.3. Homotopy Coherent Adjunctions 364.4. Homotopical uniqueness of homotopy coherent adjunctions 425. Weighted limits in qCat∞ 505.1. Weighted limits and colimits 505.2. Weighted limits in the quasi-categorical context 535.3. The collage construction 556. The formal theory of homotopy coherent monads 586.1. Weighted limits for the formal theory of monads. 596.2. Conservativity of the monadic forgetful functor 636.3. Colimit representation of algebras 657. Monadicity 717.1. Comparison with the monadic adjunction 727.2. The monadicity theorem 74References 78

1. Introduction

Quasi-categories, introduced by Boardman and Vogt [3], are now recognised as a conve-nient model for (∞, 1)-categories, i.e., categories weakly enriched over spaces. The basiccategory theory of quasi-categories has been developed by Joyal [11, 13, 12], Lurie [18, 19](under the name ∞-categories), ourselves [25], and others. Ordinary category theory canbe understood to be a special case: categories form a full subcategory of quasi-categoriesand this full inclusion respects all (quasi-)categorical definitions. As a consequence, wefind it productive to identify a category with its nerve, the corresponding quasi-category.

This paper is a continuation of [25], references to which will have the form I.x.x.x, whichdevelops the category theory of quasi-categories using 2-categorical techniques applied tothe (strict) 2-category of quasi-categories qCat2. First studied by Joyal [12], qCat2 can beunderstood to be a quotient of the simplicially enriched category qCat∞ of quasi-categories;qCat2 is defined by replacing each hom-space of qCat∞ by its homotopy category. While ourdiscussion focuses on quasi-categories, as was the case in [25] our proofs generalise withoutchange to other similar “∞-cosmoi”, subcategories of fibrant objects in model categoriesenriched over the Joyal model structure in which all fibrant objects are cofibrant. Examplesinclude complete Segal objects (“Rezk objects”) in model categories permitting Bousfieldlocalisation with sufficiently many cofibrant objects. This perspective will be exploredmore fully in the forthcoming [27].

1.1. Adjunction data. In [25], we develop the theory of adjunctions between quasi-categories, defined to be adjunctions in qCat2. Examples include adjunctions betweenordinary, topological, or locally Kan simplicial categories; simplicial Quillen adjunctions;

HOMOTOPY COHERENT ADJUNCTIONS 3

and adjunctions constructed directly on the quasi-categorical level from the existence ofappropriate limits or colimits. Explicitly, the data of an adjunction

Au

22⊥ B

frr η : idB ⇒ uf ε : fu⇒ idA

consists of two quasi-categories A,B; two functors f, u (maps of simplicial sets betweenquasi-categories); and two natural transformations η, ε represented by simplicial maps

B

i0��

Au //

i0��

B

f��

B ×∆1η// B and A×∆1 ε // A

B

i1

OO

f// A

u

OO

A

i1

OO

Because the hom-spaces BA and AB are quasi-categories, for any choices of 1-simplicesrepresenting the unit and counit, there exist 2-simplices

A×∆2 α→ B and B ×∆2 β→ A

which witness the triangle identities in the sense that their boundaries have the form

ufu

αuε

fuf

β

εf

u

ηu>>

idu

// u f

fη>>

idf

// f

(1.1.1)

This elementary definition of an adjunction between quasi-categories has a very differentform from the definition given by Lurie in [18], but they are equivalent (see I.4.4.5 for oneimplication and [27] for the converse).

This 0-, 1-, and 2-dimensional data suffices to establish that the functors f and u forman adjunction of quasi-categories, but higher dimensional adjunction data certainly exists.For example, the 1-simplices fuε and ε in AA can be composed, defining a 1-simplex wemight call εε. The 2-simplex witnessing this composition, the 2-simplex ε · σ0, and the2-simplex fα combine to form a (3, 2)-horn in AA

fufu

fuε

��

εε

""

fu

fηu<<

ε // idA

fu

ε

<<

(1.1.2)

4 RIEHL AND VERITY

which may be filled to define a 3-simplex ω and 2-simplex µ witnessing that fηu composedwith εε is ε.

An analogous construction replaces fuε with εfu and fα with βu. On account of thecommutative diagram in the homotopy category h(AA)

fufu

εfu��

fuε//

εε

""

fu

ε

��

fu ε// idA

(1.1.3)

we may choose the same 1-simplex εε as the composite of εfu with ε. Filling the (3, 2)-hornin AA

fufu

εfu

��

εε

""

fu

fηu<<

ε // idA

fu

ε

<<

(1.1.4)

produces a 3-simplex τ together with another 2-simplex witnessing that ε = εε · fηu. Butit is not immediately clear whether ω and τ may be chosen compatibly, i.e., with common2

nd face.As a consequence of our first main theorem, we will see that the answer is yes and,

furthermore, compatible choices always exist “all the way up”. To state this result werequire a new definition. To that end recall that in [28], Schanuel and Street introduce thefree adjunction: a strict 2-category Adj which has the universal property that 2-functorsAdj→ K stand in bijective correspondence to adjunctions in the 2-category K. In honourof this result, their 2-category Adj is called the free adjunction.

Inspired by their pioneering insight, we define a simplicial category, which we also callAdj, for which we prove the following result:

4.3.9,4.3.11. Theorem. Any adjunction of quasi-categories extends to a homotopy coherentadjunction: any choice of low-dimensional adjunction data for an adjunction between quasi-categories can be extended to a simplicial functor Adj→ qCat∞.

We then show, in theorems 4.4.11 and 4.4.18, that suitably defined spaces of all suchextensions are contractible. These existence and homotopy uniqueness results provide uswith the appropriate homotopy theoretic generalisation of the Schanuel-Street result to thequasi-categorical context. Consequently, we feel justified in calling our simplicial categoryAdj the free homotopy coherent adjunction.

1.2. The free homotopy coherent adjunction. We can say more about the relationshipbetween our simplicial category Adj and the Schanuel-Street free adjunction. Indeed it is asomewhat unexpected and perhaps a little remarkable fact that these two are actually one


and the same gadget. More precisely, if we look upon the Schanuel-Street free adjunctionas a simplicial category, by applying the fully faithful nerve functor to each of its hom-categories, then it is isomorphic to our free homotopy coherent adjunction. This result,which appears here as corollary 3.3.5, explains our adoption of the common notation Adjto name both of these structures.

Now observe that, as a 2-category, the hom-spaces of our simplicial category Adj are allquasi-categories, a “fibrancy” condition that indicates all possible composites of coherencedata are present in Adj and thus picked out by a simplicial functor with this domain.But of course, these hom-spaces, as nerves of categories, have unique fillers for all innerhorns, which says furthermore that this coherence data is “minimally chosen” or “maximallycoherent” in some sense.

What is unexpected from this definition, and yet essential in order to prove the “freeness”of the homotopy coherent adjunction, is that Adj is also cofibrant in the sense of being acofibrant object in the Bergner model structure on simplicial categories [1]. The cofibrantsimplicial categories are exactly the simplicial computads, which we describe in section 2.This notion has antecedents in the computads of Street [30] and the description of thecofibrant objects in the model structure described by Dwyer and Kan on the categorysimplicial categories with fixed object set [7].

In section 3, we prove that Adj is a simplicial computad by presenting an explicit simpli-cial subcomputad filtration that is then employed in the proof that any quasi-categoricaladjunction underlies a homotopy coherent adjunction. Our approach is somewhat round-about. We defined Adj first as a simplicial category by introducing a graphical calculusfor its n-arrows. This graphical calculus seamlessly encodes all the necessary simplicialstructure, while highlighting a set of atomic arrows that freely generate the arrows in eachdimension under horizontal composition. We then prove that this simplicial category isisomorphic to the 2-category Adj under the embedding 2-Cat ↪→ sSet-Cat.

In section 4, we prove that any adjunction of quasi-categories extends to a homotopycoherent adjunction and moreover that homotopy coherent adjunctions Adj → qCat∞extending a given adjunction of quasi-categories are “homotopically unique”. In fact, as ouruse of the subcomputad filtration of Adj makes clear, there are many extension theorems,distinguished by what we take to be the initial data of the adjunction of quasi-categories.We define spaces of extensions from a single left adjoint functor; from choices of bothadjoints and a representative for the counit; from choices of both adjoints, representativesfor the unit and counit, and a representative for one of the triangle identities; and so on,proving that each of these defines a contractible Kan complex.

1.3. Weighted limits and the formal theory of monads. The 2-category Adj hastwo objects, which we denote “+” and “−”; their images specify the objects spanned bythe adjunction. The hom-category Adj(+,+) is �+—the “algebraist’s delta”—the categoryof finite ordinals and order-preserving maps. Ordinal sum makes �+ a strict monoidalcategory; indeed, it is the free strict monoidal category containing a monoid. Hence, a2-functor whose domain is the one-object 2-category with hom-category �+ is exactly amonad in the target 2-category. The hom-category Adj(−,−) is �op

+ . In this way, the

6 RIEHL AND VERITY

restrictions of the free adjunction Adj to the subcategories spanned by one endpoint or theother define the free monad and the free comonad.

Restrictions of a homotopy coherent adjunction to the subcategories Mnd and Cmdspanned by + and − respectively define a homotopy coherent monad and a homotopycoherent comonad . Unlike the case for adjunctions, (co)monads in qCat2 are not auto-matically homotopy coherent; a monad is an algebraically-defined structure whereas anadjunction encodes a universal property. However, as a corollary of our extension theorem,any monad arising from an adjunction extends to a homotopy coherent monad, a simplicialfunctor with domain Mnd ↪→ Adj. In the body of this paper, except when discussing mon-ads, we frequently omit the appellation “homotopy coherent” because in the other settingsthis interpretation is automatic: categories regarded as quasi-categories via their nervesautomatically define homotopy coherent diagrams.

In the second half of this paper, we present a “formal” re-proof of the quasi-categoricalmonadicity theorem that also illuminates the classical categorical argument. The startinginsight is a characterisation of the quasi-category of algebras for a homotopy coherentmonad as a weighted limit. In this context, a weight is a functor describing the “shape” of ageneralised cone over a diagram indexed by a fixed small category. An object representingthe set of cones described by a particular weight is called a weighted limit. The use ofweighted limits can provide a useful conceptual simplification because calculations involvingthe weights reveal the reason why these results are true; cf. the expository paper [24].

To make use of weighted limits in the quasi-categorical context a preliminary result isneeded because the simplicial subcategory qCat∞ ↪→ sSet is not complete. It is howeverclosed under weighted limits whose weights are cofibrant in the projective model structureon the appropriate sSet-valued diagram category. We prove this result and provide a generalreview of the theory of weighted limits in section 5. We anticipate other uses of the factthat quasi-categories are closed under weighted limits with projectively cofibrant weightsthan those given here. In a supplemental paper [26], we prove that for any diagram ofquasi-categories admitting (co)limits of shape X and functors that preserve these colimits,the weighted limit again admits (co)limits of shape X.

In section 6, we define the quasi-category of algebras B[t] associated to a homotopy coher-ent monad t on a quasi-category B as a limit weighted by the restriction along Mnd ↪→ Adjof the covariant simplicial functor represented by the object −. The homotopy coherentmonadic adjunction f t a ut : B[t]→ B is then defined formally: it is simply a reflection inqCat∞ of an adjunction between the weights whose limits identify the two quasi-categoriesinvolved. In particular, the monadic forgetful functor ut is induced from a natural transfor-mation between weights that is a projective cofibration and “constant on dimension zero”in an appropriate sense. It follows that the induced map of weighted limits is conservative(reflects isomorphisms).

We give an explicit description of the vertices in the quasi-category of algebras for a ho-motopy coherent monad, unpacking the weighted limit formula. A calculation on weights—reminiscent of our proof in [25] that for any simplicial object in a quasi-category admittingan augmentation and a splitting the augmentation defines the colimit—proves that thesevertices are “codescent objects”.


6.3.17.Theorem. Any vertex in the quasi-category B[t] of algebras for a homotopy coherentmonad is the colimit of a canonical ut-split simplicial object of free algebras.

In section 7, we compare a general homotopy coherent adjunction extending f a u :A → B with the induced monadic adjunction defined from its homotopy coherent monadt = uf . A map between weights, this time indexed on the simplicial category Adj, inducesa canonical simplicial natural transformation from the homotopy coherent adjunction tothe monadic adjunction. The monadicity theorem gives conditions under which the non-identity component of this map is an equivalence of quasi-categories.

7.2.4,7.2.7. Theorem. There is a canonical comparison functor defining the component ofa simplicial natural transformation between any homotopy coherent adjunction f a u andits monadic homotopy coherent adjunction f t a ut.

A //

u

��

B[t]

ut}}B

f

__f t==

If A admits colimits of u-split simplicial objects, then the comparison functor admits a leftadjoint. If u preserves colimits of u-split simplicial objects and reflects isomorphisms, thenthis adjunction defines an adjoint equivalence A ' B[t].

1.4. Acknowledgments. During the preparation of this work the authors were supportedby DARPA through the AFOSR grant number HR0011-10-1-0054-DOD35CAP and by theAustralian Research Council through Discovery grant number DP1094883. The first-namedauthor was also supported by an NSF postdoctoral research fellowship DMS-1103790.

We are grateful to Dimitri Zaganidis, who noticed an error in a lemma used in our firstproof of theorem 6.3.17. We would also like to thank Mike Hopkins for his continuedsupport and encouragement.

2. Simplicial computads

Cofibrant simplicial categories are simplicial computads, a definition we introduce in§2.1 together with some important examples. The notion of simplicial computad providesa direct characterisation of those simplicial categories that are cofibrant that is useful forinductive arguments: a simplicial functor whose domain is a simplicial computad is definedby specifying images of the atomic non-degenerate n-arrows. In §2.2, we study simplicialsubcomputads in order to describe what will be needed for the “induction steps” in theproofs of section 4, which require extensions along simplicial subcomputad inclusions.

2.1. Simplicial categories and simplicial computads.

2.1.1. Notation (simplicial categories). It will be convenient to identify simplicially en-riched categories, simplicial categories henceforth, as simplicial objects in Cat. The cate-gory of simplicial categories is isomorphic to the full subcategory of Cat�

opof those sim-

plicial objects A : �op → Cat for which the simplicial set obtained by composing with the

8 RIEHL AND VERITY

object functor obj : Cat → Set is constant. In other words, a simplicial object in Cat isa simplicial category just when each of the categories in the diagram has the same set ofobjects and each of the functors is the identity on objects.

Given a simplicial category A : �op → Cat, an n-arrow is an arrow in An; an n-arrowf : a → b is precisely an n-simplex in the simplicial set A(a, b). We write ∅ for the initialsimplicial category on no objects and 1 for the terminal simplicial category on a singleobject. We adopt the same terminology for large simplicial categories K, using the sizeconventions detailed in I.2.0.1.

2.1.2. Notation (whiskering in a simplicial category). For each vertex in a simplicial setand for each n > 0, there is a unique degenerate n-simplex on that vertex obtained byacting via the simplicial operator [n] → [0]. If f : a → b is an n-arrow in a simplicialcategory A, and x : a′ → a and y : b→ b′ are 0-arrows, we write fx : a′ → b and yf : a→ b′

for the n-arrows obtained by degenerating x and y and composing in A. We refer to thisoperation as whiskering the n-arrow f with x or y; in the special case where the simplicialcategory is a 2-category, this coincides with the usual notion.

2.1.3. Example (the generic n-arrow). For any simplicial set X, let 2[X] denote thesimplicial category with two objects 0 and 1 and whose only non-trivial hom-space is2[X](0, 1) := X. Here we define 2[X](1, 0) = ∅ and 2[X](0, 0) = 2[X](1, 1) = ∗.

For any simplicial category K, a simplicial functor F : 2[X] → K is completely deter-mined by the following data:• a pair of objects B and A in K and• a simplicial map f : X → K(B,A).

On account of the canonical bijection between simplicial functors 2[∆n]→ K and n-arrowsof K, we refer to the simplicial category 2[∆n] as the generic n-arrow.

2.1.4. Definition ((relative) simplicial computads). The class of relative simplicial com-putads is the class of all simplicial functors which can be expressed as a transfinite com-posite of pushouts of coproducts of• the unique simplicial functor ∅ ↪→ 1, and• the inclusion simplicial functor 2[∂∆n] ↪→ 2[∆n] for n ≥ 0.

A simplicial category A is a simplicial computad if and only if the unique functor ∅ ↪→ Ais a relative simplicial computad.

2.1.5. Observation (an explicit characterisation of simplicial computads). An arrow f inan unenriched category is atomic if it is not an identity and it admits no non-trivialfactorisations, i.e., if whenever f = g ◦ h then one or other of g and h is an identity. Acategory is freely generated (by a reflexive directed graph) if and only if each of its non-identity arrows may be uniquely expressed as a composite of atomic arrows. In this case,the generating graph is precisely the subgraph of atomic arrows.

An extension of this kind of characterisation gives an explicit description of the simplicialcomputads. Specifically, A is a simplicial computad if and only if:


• each non-identity n-arrow f of An may be expressed uniquely as a composite f1 ◦ f2 ◦· · · ◦ f` in which each fi is atomic, and• if f is an atomic n-arrow in An and α : [m] → [n] is a degeneracy operator in � thenthe degenerated m-arrow f · α is atomic in Am.

On combining this characterisation with the Eilenberg-Zilber lemma, we find that A is asimplicial computad if and only if all of its non-identity arrows f can be expressed uniquelyas a composite

f = (f1 · α1) ◦ (f2 · α2) ◦ · · · ◦ (f` · α`) (2.1.6)in which each fi is non-degenerate and atomic and each αi ∈ � is a degeneracy operator.

2.1.7. Observation. A simplicial functor is called a trivial fibration of simplicial categoriesif it has the right lifting property with respect to the generating set of simplicial functorsof definition 2.1.4. A simplicial functor P : E → B is a trivial fibration if and only ifit is surjective on objects and its action E(A,B) → B(PA, PB) on each hom-space isa trivial fibration of simplicial sets. A simplicial functor is said to be a cofibration ofsimplicial categories if it is a retract of a relative simplicial computad. These are theclasses appearing in Bergner’s model structure on simplicial categories [1].

The characterisation of observation 2.1.5 reveals that all retracts of simplicial computadsare again simplicial computads and hence that the cofibrant objects in Bergner’s modelstructure are precisely the simplicial computads: no retracts are needed.

2.1.8. Example. The simplicial categories 2[X] defined in 2.1.3 are simplicial computads,with every simplex in X an atomic arrow.

2.1.9. Example (free simplicial resolutions define simplicial computads). There is a free-forgetful adjunction

CatU

22⊥ GphF

rr

between small categories and reflexive directed graphs inducing a comonad FU on Cat.The comonad resolution associated to a small category C is a simplicial computad FU•C

FUC FηU // FUFUCFUεoo

εFUoo

FUFηU //

FηUFU //

FUFUFUC · · ·FUεFUoo

εFUFUoo

FUFUεoo

called the standard resolution of C in [7]. The category FUC is the free category on theunderlying graph of C. Its arrows are (possibly empty) strings of composable non-identityarrows of C. The atomic 0-arrows are the non-identity arrows of C. An n-arrow is a stringof composable arrows in C with each arrow in the string enclosed in exactly n pairs ofparentheses. The atomic n-arrows are those strings enclosed in a single pair of “outermost”parentheses.

10 RIEHL AND VERITY

2.1.10. Example. The simplicial computad FU•C is isomorphic to the image of the nerveof C under the left adjoint to the homotopy coherent nerve

sSet-CatN

22⊥ sSetC

rr

cf. [21, 6.7]. Indeed, for any simplicial set X, CX is a simplicial computad. This followsfrom the fact that C a N defines a Quillen equivalence between the model structures ofBergner and Joyal, but we prefer to give a direct proof.

The arrows in CX admit a simple geometric characterisation due to Dugger and Spivak[5]: n-arrows in CX are necklaces in X, i.e., maps ∆n1 ∨ ∆n2 ∨ · · · ∨ ∆nk → X from asequence of standard simplices joined head-to-tail, together with a nested sequence of n−1sets of vertices. The atomic arrows are precisely those whose necklace consists of a singlesimplex.

2.1.11. Example. In particular, the simplicial category C∆n whose objects are integers0, 1, . . . , n and whose hom-spaces are the cubes

C∆n(i, j) =

(∆1)j−i−1 i < j

∆0 i = j

∅ i > j

is a simplicial computad. In each hom-space, the atomic arrows are precisely those whosesimplices contain the initial vertex in the poset whose nerve defines the simplicial cube.

2.2. Simplicial subcomputads. The utility of the notion of simplicial computad is thefollowing: ifA is a simplicial computad andK is any simplicial category, a simplicial functorA → K can be defined inductively simply by specifying images for the non-degenerate,atomic n-arrows in a way that is compatible with previously chosen faces. Let us nowmake this idea precise.

2.2.1.Definition (simplicial subcomputad). IfA is a simplicial computad then a simplicialsubcomputad B of A is a simplicial subcategory that is closed under factorisations: i.e.,• if g and f are composable arrows in A and g ◦ f is in B, then both g and f are in B.

This condition is equivalent to postulating that B is a simplicial computad and that everyarrow which is atomic in B is also atomic in A.

2.2.2. Observation (simplicial subcomputads and relative simplicial computads). If B is asimplicial subcomputad of the simplicial computad A, then the inclusion functor B ↪→ A isa relative simplicial computad. Indeed, every relative simplicial computad may be obtainedas a composite of pushouts of simplicial subcomputad inclusion. Furthermore, if C is asimplicial subcategory of B then C is a simplicial subcomputad of B if and only if it is asimplicial subcomputad of A.

2.2.3. Example. If X is a simplicial subset of Y , then 2[X] is a simplicial subcomputadof 2[Y ], and CX is a simplicial subcomputad of CY .


2.2.4. Definition. The simplicial subcomputad generated by a set of arrows X in a simpli-cial computad A is the intersection X of all of the simplicial subcomputads of A whichcontain X. Note that arbitrary intersections of simplicial subcomputads are again simpli-cial subcomputads. The simplicial subcomputad X can be formed by inductively closingX up to the smallest subset of A containing it which satisfies the closure properties:• if f ∈ X and α is a simplicial operator then f · α ∈ X, and• if g ◦ f is a composite in A, then g ◦ f ∈ X if and only if g and f are both in X.

2.2.5. Example ((co)skeleta of simplicial categories). The r-skeleton skrA (r ≥ −1) ofa simplicial category A is the smallest simplicial subcategory of A which contains all ofits arrows of dimension less than or equal to r. We say that a simplicial category A isr-skeletal if skrA = A, i.e., when all of its arrows of dimension greater than r can beexpressed as composites of degenerate arrows. When A is a simplicial computad, an arrowf is in skr(A) if and only if each arrow fi in the decomposition of (2.1.6) has dimension atmost r. In this case, the skeleton skrA is the simplicial subcomputad of A generated byits set of r-arrows. By convention, we write sk−1 A for the discrete simplicial subcategorywhich contains all of the objects of A.

Each r-skeleton functor has a right adjoint

sSet-Catcoskr

22⊥ sSet-Catskr

rr

which as ever we call the r-coskeleton. The 0-coskeleton cosk−1 A is the chaotic simplicialcategory on the objects of A. The r-coskeleton of A is defined by applying the usualsimplicial r-coskeleton functor coskr : sSet → sSet to each hom-space A(A,B). The con-sequent hom-spaces (coskrA)(A,B) := coskr(A(A,B)) inherit a compositional structurefrom that of A by dint of the fact that the simplicial r-coskeleton functor is right adjointand thus preserves all finite products. A simplicial category A is r-coskeletal if and onlyif the adjoint transpose A → coskrA of the inclusion skrA ↪→ A is an isomorphism. Soa simplicial category is (−1)-coskeletal when all of its hom-spaces are isomorphic to theone point simplicial set ∆0 and it is r-coskeletal precisely when each of its hom-spaces isr-coskeletal in the usual sense for simplicial sets.

2.2.6. Proposition. If A is a simplicial computad, a simplicial functor F : A → K isuniquely specified by choosing• an object F (A) in K for each of the objects A of A, and• an arrow F (f) in K for each of the non-degenerate and atomic arrows f of A

subject to the conditions that• these choices are made compatibly with the dimension, domain, and codomain operationsof A and K, and• whenever f is a non-degenerate and atomic arrow and its face f · δi is decomposed interms of non-degenerate and atomic arrows fi as in (2.1.6) then the face F (f) · δi isthe corresponding composite of degenerate images of the F (fi).

12 RIEHL AND VERITY

Proof. We induct over the skeleta of A. The specification of objects defines F : sk−1 A→K. If F : skr−1 A→ K is a simplicial functor, then extensions of F to a simplicial functorskrA→ K are uniquely specified by the following data:• an r-arrow F (f) in K for each atomic non-degenerate r-arrow f in A subject to thecondition that F (f) · δi = F (f · δi) for each elementary face operator δi : [r − 1] → [r]in �.

This condition on the faces of the chosen arrow F (f) makes sense because the faces f · δiof any r-arrow are (r− 1)-arrows and are thus elements of skr−1 A to which we may applythe un-extended simplicial functor F : skr−1 A→ K.

To construct a simplicial functor from this data we simply decompose each arrow f ofskr(A) as in (2.1.6) and then observe that F (fi) is defined for each component of thatdecomposition either because fi is in skr−1 A or because it is a non-degenerate and atomicr-arrow and thus has an image in K given by the extra data supplied above. This thenprovides us with a value for F (f) given by the composite

F (f) := (F (f1) · α1) ◦ (F (f2) · α2) ◦ · · · ◦ (F (f`) · α`) (2.2.7)and it is easily checked, using the uniqueness of these decompositions in A, that this actionis functorial and that it respects simplicial actions. In other words, this result tells us thatwe may build the skeleton skr(A) from the skeleton skr−1(A) by glueing on copies of thecategory 2[∆r] along functors 2[∂∆r] → skr−1(A), one for each atomic non-degenerater-arrow of A. �

3. The generic adjunction

In this section, we introduce a simplicial category Adj via a graphical calculus developedin §3.1, from which definition it will be immediately clear that we have defined a simplicialcomputad. This result, when combined with proposition 2.2.6, will make it relativelyeasy to construct simplicial functors whose domain is Adj. In §3.2 and §3.3, we thenshow that Adj is isomorphic to the simplicial category obtained by applying the nerveto each hom-category in the free 2-category containing an adjunction [28]. To emphasisethe interplay between 2-categories and simplicial categories, an important theme of ourwork, our proof strategy is somewhat indirect. In §3.2, we show that the hom-spacesof Adj satisfy the Segal condition; thus Adj is isomorphic to some 2-category under theembedding 2-Cat ↪→ sSet-Cat. In §3.3, we show that Adj has the same universal propertyas the Schanuel and Street 2-category, proving that these gadgets are isomorphic andjustifying our decision not to notationally distinguish between them.

3.1. A graphical calculus for the simplicial category Adj. To define a small simplicialcategory, thought of as an identity-on-objects simplicial object in Cat, it suffices to specify• a set of objects,• for each n ≥ 0, a set of n-arrows with (co)domains among the specified object set,• a right action of the morphisms in � on this graded set,• a “horizontal” composition operation for n-arrows with compatible (co)domains thatpreserves the simplicial action.


We will define Adj to be the simplicial category with two objects, denoted “+” and “−”,and whose n-arrows will be certain graphically inspired strictly undulating squiggles onn+ 1 lines. We will provide a formal account of these squiggles presently, but we prefer tostart by engaging the reader’s intuition with a picture. For example, the diagram belowdepicts a 6-arrow in the hom-space Adj(−,+):

6

5

4

3

2

1

−

+

0

1

2

3

4

5

6

+, 2, 6, 3, 4, 1,+, 2, 3,−

(3.1.1)

Here we have drawn the n + 1 lines (n = 6 in this case) which support this squiggle ashorizontal dotted lines numbered 0 to n down the right hand side, and these lines separaten + 2 levels which are labelled down the left hand side. The levels which sit between apair of lines, sometimes called gaps, are labelled 1 to n while the top and bottom levels arelabelled − and + respectively.

Each turning point of the squiggle itself lies entirely within a single level. The qualifier“strict undulation” refers to the requirement that the levels of adjacent turning pointsshould be distinct and that they should oscillate as we proceed from left to right. Forexample, the following is not a strictly undulating squiggle

3

2

1

−

+

0

1

2

3

−, 2, 1, 3, 3,+because its last two turning points occur on the same level.

The data of such a squiggle can be encoded by a string a = (a0, a1, . . . , ar) of letters in theset {−, 1, 2, . . . , n,+}, corresponding to the levels of each successive turning point, subjectto conditions that we will enumerate shortly. The string corresponding to our 6-arrow(3.1.1) is displayed along the bottom of that picture. As we shall see, composition of n-arrows in Adj will correspond to a coalesced concatenation operation on these strings, andso it is natural to read them from right to left. Consequently, the domain and codomain ofsuch a squiggle are naturally taken to be its last and first letters respectively; in particular,the domain of the 6-arrow (3.1.1) is − and its codomain is +.

14 RIEHL AND VERITY

3.1.2. Definition (strictly undulating squiggles). We write a = (a0, a1, . . . , ar) for a non-empty string of letters in {−, 1, 2, ..., n,+}, intended to represent a squiggle on n+ 1 lines,with domain dom(a) := ar and codomain cod(a) := a0. We define the width of this squiggleto be the number w(a) := r, that is, the number of letters in its string minus 1. The interiorof such a string is the sub-list a1, ..., ar−1 of all of its letters except for those at its end pointsa0 and ar.

We say that a string a represents a strictly undulating squiggle on n+1 lines if it satisfiesthe conditions that:

(i) a0, aw(a) ∈ {−,+}, and(ii) if a0 = − (resp. a0 = +) then for all 0 ≤ i < w(a) we have ai < ai+1 whenever i is

even (resp. odd) and ai > ai+1 whenever i is odd (resp. even).We also say that a is simply an undulating squiggle on n+1 lines if it satisfies condition (i)above but only satisfies the weaker condition

(ii)′ if a0 = − (resp. a0 = +) then for all 0 ≤ i < w(a) we have ai ≤ ai+1 whenever iis even (resp. odd) and ai ≥ ai+1 whenever i is odd (resp. even).

in place of condition (ii).

3.1.3. Definition (composing squiggles). Two such n-arrows b and a are composable whenbw(b) = a0 and their composite is described graphically as the kind of horizontal glueingdepicted in the following picture:

4

3

2

1

−

+

0

1

2

3

4

−, 2, 1, 4, 1, 3,−

◦4

3

2

1

−

+

0

1

2

3

4

−, 4, 1, 3, 2,+

=

4

3

2

1

−

+

0

1

2

3

4

−, 2, 1, 4, 1, 3,−, 4, 1, 3, 2,+More formally, the composite b ◦ a is given by the string (b0, · · · , bw(b) = a0, a1, · · · , aw(a))constructed by dropping the last letter of b and concatenating the resulting string with a.It is easily seen that this composition operation is associative and that it has the n-arrows(−) and (+) as identities. In other words, these operations make the collection of n-arrowsinto a category with objects − and +.

3.1.4. Observation (atomic n-arrows). Notice that an n-arrow c of Adj may be expressed asa composite b ◦ a of non-identity n-arrows precisely when there is some 0 < k < w(c) suchthat ck ∈ {−,+}. Specifically, b := (c0, ..., ck) and a := (ck, ..., cw(c)) are strictly undulatingsquiggles whose composite is c. It follows that an n-arrow c is atomic, in the sense ofdefinition 2.1.4, if and only if the letters − and + do not appear in its interior.

We now describe how the simplicial operators act on the arrows of Adj.

3.1.5. Observation (simplicial action on strictly undulating squiggles). The geometric ideabehind the simplicial action is simple: given a simplicial operator α : [m] → [n] and a


strictly undulating squiggle on n + 1 lines, one produces a strictly undulating squiggle onm + 1 lines by removing the line labelled by each letter i ∈ [n] not in the image of α,“pulling” the string taught if necessary to preserve the strict undulations, and replacingthe line labeled by each letter i ∈ [n] that is in the image by an identical copy of that linefor each element of the fiber α−1(i), “stretching” apart these lines to create the appropriategaps.

To illustrate, consider the 4-arrow

a =

4

3

2

1

−

+

0

1

2

3

4

−, 2, 1, 4, 1, 3,−

(3.1.6)

Its image under the action of the degeneracy operator σ0 : [5]→ [4] is the 5-arrow

a · σ0 =

5

4

3

2

1

−

+

0

1

2

3

4

5

−, 3, 2, 5, 2, 4,−obtained by doubling up line 0 and opening up an extra gap between these copies. Fromthis description, it is clear that degenerate n-arrows are readily identifiable: an n-arrow isin the image of the degeneracy operator σi if and only if the letter i + 1 does not appearin its representing string.

The image of a under the action of the face operator δ4 : [3] → [4] is the 3-arrow con-structed by removing the line numbered 4 from the squiggle (3.1.6):

a · δ4 =3

2

1

−

+

0

1

2

3

−, 2, 1,+, 1, 3,−This is again a strictly undulating squiggle which we may immediately take to be the facewe seek. Note that a · δ4 is decomposable even though a itself is atomic.

16 RIEHL AND VERITY

The construction of the image of a under the action of the face operator δ1 : [3] → [4]must be constructed in two steps. First we remove the line numbered 1 from the squigglein (3.1.6) to give:

3

2

1

−

+

0

1

2

3

−, 1, 1, 3, 1, 2,−This, however, isn’t strictly undulating so we eliminate matched pairs of those adjacentturning points which occur at the same level to give the desired 3-face:

a · δ1 =3

2

1

−

+

0

1

2

3

−, 3, 1, 2,−Note that the execution of this reduction step means that the width of this particular faceis strictly less than that of the original n-arrow.

In general, the elementary face operator δi acts on an n-arrow a of Adj by first replacingeach letter aj > i with aj−1 and then, if necessary, performing a reduction step to preservethe strict undulation. This reduction deletes consecutive matched pairs of repetitions ofthe same letter, eliminating sequences of repetitions of even length entirely and reducingthose of odd length to a single letter. It is now straightforward to show that the stringthat emerges from this reduction step will be strictly undulating and thus will deliver usan (n− 1)-arrow of Adj.

Applying this algorithm to take further faces we see, for example, that the image of aunder the face operator {0, 1, 4} : [2]→ [4] is the 2-arrow

a · δ3δ2 = a · δ2δ2 = 2

1

−

+

0

1

2

−, 2, 1, 2, 1, 2,−


and in turn the first face of this 2-arrow is the 1-arrow

1

−

+

0

1

−, 1, 1, 1, 1, 1,−

reduce

1

−

+

0

1

−, 1,−A formal account of the right action of � on the arrows of Adj makes use of the interval

representation.

3.1.7. Observation (interval representation). There is a faithful “interval representation” inthe form of a functor ir : �op ↪→ � defined on objects by ir([n]) := [n+1] and on morphismsby:• for any elementary face operator δin : [n− 1]→ [n],

ir(δin) := σin : [n+ 1]→ [n],

and• for any elementary degeneracy operator σin−1 : [n]→ [n− 1],

ir(σin−1) := δi+1n+1 : [n]→ [n+ 1].

This functor is faithful and maps onto the full subcategory of those simplicial operatorswhich preserve distinct top and bottom elements. Following Joyal [10], who calls thesubcategory of top and bottom preserving maps in � the category of intervals, we refer tothe image of �op in � as the category of strict intervals.

We can think of the numbers labelling the lines on the right of our squiggle diagrams asbeing the elements of an ordinal [n] in �op and those labelling the levels on the left as beingthe elements of the corresponding ordinal [n+ 1] = ir([n]) in �. Here we have renamed thetop and bottom elements of [n + 1] to “+” and “−” respectively in order to indicate theirspecial status in the category of strict intervals. From this perspective, we may motivatethis action of the interval representation on elementary simplicial operators using the linesand levels of our squiggle diagrams:• The action of an elementary face operator δi : [n − 1] → [n] on a squiggle diagramproceeds by removing the line labelled i on the right, which causes the levels labelled iand i+ 1 on the left to be coalesced into one and causes all higher numbered levels tohave their level number reduced by one. This operation maps levels labelled in [n+ 1]to levels labelled in [n] according to the action of the elementary degeneracy operatorir(δi) = σi : [n+ 1]→ [n].• The action of an elementary degeneracy operator σi : [n]→ [n−1] on a squiggle diagramproceeds by doubling up on the line labelled i on the right and then stretching apartthose lines to give a new level labelled i + 1 on the left, which causes the levels whoselabels are greater than i to have their level numbers increased by one. This operationmaps levels labelled in [n] to levels labelled in [n + 1] according to the action of theelementary face operator ir(σi) = δi+1 : [n]→ [n+ 1].

18 RIEHL AND VERITY

3.1.8.Definition (simplicial actions formally). The action of a simplicial operator α : [m]→[n] on an n-arrow of Adj may be formally described via a two-step process:• Apply the interval representation ir(α) : [n + 1] → [m + 1] of α to the entries of a =

(a0, ..., ar) to give a string (a′0, ..., a′r). This string is not necessarily strictly undulating,

since ir(α) may not be injective, but it is undulating.• Reduce the undulating string (a′0, ..., a

′r) to a strictly undulating one by iteratively

locating consecutive pairs of matched letters and eliminating them. The order of theseeliminations is immaterial and that this process will always terminate at the samestrictly undulating string, which we take to be a · α.The actions of the simplicial operators on the arrows of Adj respect the simplicial iden-

tities and are compatible with composition and thus make Adj into a simplicial category.For aesthetic reasons, and because their data is redundant in the presence of an orientedplanar picture, when drawing squiggles we often decline to include the labels for the linesand gaps, or the corresponding string of letters.

3.1.9. Observation (the vertices of an arrow in Adj). The 2nd vertex of the 4-arrow (3.1.6)can be computed by deleting all lines except for the one labelled 2 and reducing the resultingundulating squiggle to a strictly undulating one:

a =

0

1

2

3

4

−, 2, 1, 4, 1, 3,−

delete

−,−,−,+,−,+,−

reduce a · {2} =

−,+,−,+,−

More directly, the diagram of the jth vertex of a is the squiggle on one line crossing thesame number of times that the original squiggle crosses the line j. Writing {j} for thesimplicial operator that picks out the jth vertex, we have:

a · {0} =

−,+,−a · {1} =

−,+,−,+,−,+,−a · {3} =

−,+,−a · {4} =

−The last of these vertices, denoted by an empty picture, is the identity 0-arrow on theobject −.

A key advantage to our explicit description of Adj is that the proof of the followingimportant proposition is trivial.

3.1.10. Proposition. The simplicial category Adj is a simplicial computad.


Proof. A squiggle in Adj may be uniquely decomposed into a sequence of atomic arrowsby splitting it at each successive − or + letter in its interior.

+, 2, 6,−, 5, 3, 4, 1,+, 2, 3,−

+, 2, 6,−

◦

−, 5, 3, 4, 1,+

◦

+, 2, 3,−The operation of degenerating an arrow does not introduce any extra + or − letters intoits interior, from which it follows that degenerated atomic arrows are again atomic. �

3.1.11. Example (adjunction data in Adj). For later use, we name some of the low dimen-sional non-degenerate atomic arrows in Adj. There are exactly two non-degenerate atomic0-arrows in Adj, these being:

f := and u :=

Since Adj is a simplicial computad, all of its other 0-arrows may be obtained as a uniquealternating composite of those two, for example:

fufuf =

One convenient aspect of our string notation for arrows is that the act of whiskering ann-arrow a with one the arrows f or u, as described in notation 2.1.2, simply amounts toappending or prepending one of the symbols − or + as follows:

fa = a with − prepended, af = a with + appended,ua = a with + prepended, and au = a with − appended.

There are also exactly two non-degenerate atomic 1-arrows in Adj, these being:

η := and ε :=

Writing these 1-arrows as if they were 1-cells in a 2-category, they clearly take a formreminiscent of the unit η : id− ⇒ uf and counit ε : fu⇒ id+ of an adjunction. Here again,since Adj is a simplicial computad all of its 1-arrows are uniquely expressible as a composite

20 RIEHL AND VERITY

of the 1-arrows ε and η and the degenerated 1-arrows obtained from the 0-arrows u and fsuch as:

ufη = and fηηuε =

By way of contrast, there exists a countably infinite number of non-degenerate atomic2-arrows. Key amongst these are a pair of 2-arrows whose squiggle diagrams should re-mind the reader of the string diagram renditions of the familiar triangle identities of anadjunction:

α :=faces α · δ2 = ηu = , α · δ1 = u = , α · δ0 = uε =

β :=faces β · δ2 = fη = , β · δ1 = f = , β · δ0 = εf =

We can arrange all of the non-degenerate atomic 2-arrows of Adj into two countable familiesα(n) and β(n) for n ≥ 2. An arrow α(n) in the first of these families starts at +, alternatesbetween 1 and 2 for n steps and then finishes at − if n is even and at + if n is odd. Soα(2) = α and the next three elements in this sequence are:

α(3) = , α(4) = , α(5) =

The faces of the arrows in this family are given by the formulae

α(2r) · δ2 = ηru α(2r) · δ1 = u α(2r) · δ0 = uεr

α(2r+1) · δ2 = ηr+1 α(2r+1) · δ1 = η α(2r+1) · δ0 = uεrf(3.1.12)

where the expressions εr and ηr denote r-fold compositional powers of the endo-arrows εand η. The family β(n) is that obtained by reflecting the corresponding arrows in the familyα(n) through a horizontal axis.

The pair of 3-arrows discussed in section 1.1 of the introduction are:

ω := τ := µ := ω · δ2 = τ · δ2 =


3.2. The simplicial category Adj as a 2-category. Our blanket identification of cat-egories with their nerves leads to a corresponding identification of 2-categories with sim-plicial categories, obtained by applying the nerve functor hom-wise. In this section, wewill show that the simplicial category Adj is a 2-category in this sense, i.e., that its hom-spaces Adj are nerves of categories, a “fibrancy” result. Indeed, we show in §3.3 that Adjis isomorphic to the generic or walking adjunction, the 2-category freely generated by anadjunction.

Using the graphical calculus, it is not difficult to sketch a direct proof that Adj isisomorphic to the generic adjunction, whose concrete description recalled in remark 3.3.8below. However, we find it more illuminating to first verify that the hom-spaces in Adjsatisfy the Segal condition, showing that Adj is isomorphic to some 2-category, and thenprove that this 2-category has the universal property that defines the walking adjunction.

3.2.1. Recall (Segal condition). A simplicial set X is the nerve of a category if and only ifit satisfies the (strict) Segal condition, which states that for all n,m ≥ 1 the commutativesquare

Xn+m

−·{0,...,n}//

−·{n,...,n+m}��

Xn

−·{n}��

Xm −·{0}// X0

is a pullback. This condition says that if x is an n-simplex and y is an m-simplex in X forwhich the last vertex x · {n} of x is equal to the first vertex y · {0} of y then there exists aunique (n+m)-simplex z for which z · {0, ..., n} = x and z · {n, ...,m+ n} = y.

3.2.2. Proposition. Each hom-space of the simplicial category Adj is the nerve of a cate-gory.

Proof. To prove proposition 3.2.2, it suffices to verify that the arrows in each hom-space ofAdj satisfy the Segal condition. We convey the intuition with a specific example providedby the following pair of squiggles in the hom-space Adj(+,+):

a = 2

1

−

+

0

1

2

+, 1, 2,−,+, 2,+, 1, 2, 1,+

b = 2

1

−

+

0

1

2

+, 1, 2,−, 2,−, 2, 1,+,−,+

(3.2.3)

Counting the crossings of the bottom line in the first of these and the crossings of the topline in the second, as discussed in observation 3.1.9, we see that the last vertex of a is thesame as the first vertex of b. Thus, a and b are a pair of arrows to which premise of theSegal condition applies.

Since these crossings match up we may “splice” these two squiggles together by identifyingthe bottom line of a and the top line of b and then fusing each string which passes through

22 RIEHL AND VERITY

the bottom line of a with the corresponding string which passes through the top line of bto construct the following squiggle

c =

4

3

2

1

−

+

0

1

2

3

4

+, 3, 4, 1, 2,−, 4, 2, 4, 3,+, 1, 2, 1,+

(3.2.4)

To evaluate the face {0, 1, 2} : [2] → [4] of this spliced simplex, we remove lines 3 and 4and reduce

+,+,+, 1, 2,−,+, 2,+,+,+, 1, 2, 1,+

reduce

+, 1, 2,−,+, 2,+, 1, 2, 1,+

= a

which gives us back a. Similarly, to evaluate the face {2, 3, 4} : [2] → [4], we remove lines0 and 1 and reduce

+, 1, 2,−,−,−, 2,−, 2, 1,+,−,−,−,+

reduce

+, 1, 2,−, 2,−, 2, 1,+,−,+

= b

which gives us back b. Thus, c is the clearly unique 4-simplex which has c · {0, 1, 2} = aand c · {2, 3, 4} = b as required.

It is an entirely straightforward, if tedious, combinatorial exercise to express this splicingoperation as a function on the strings that encode the strictly undulating squiggles, definingan inverse to the function from the set of (n + m)-arrows of Adj to the set of pairs of n-arrows and m-arrows with common last and first vertices. Full details are given in [23,§4.2]. �

3.2.5. Remark. The simplicial category Adj turns out to be isomorphic to the Dwyer-Kanhammock localisation [6] of the category consisting of two objects and a single non-identityarrow w : +→ −, which is a weak equivalence. This was first observed by Karol Szumiło.


We give a sketch of the proof employing our graphical calculus. Consider an k-arrow

+ + + + + + + −+ + + − + − + −+ − + − + − − −+ − − − + − − −

(3.2.6)

and draw vertical lines bisecting each undulation point (but not otherwise intersecting thesquiggle) and also one vertical line to the left and to the right of the diagram. Label eachintersection of a vertical and horizontal line in this picture with a “+” if it is in the region“above” the squiggle and a “−” if it is “below”; the squiggle on the left of (3.2.6) gives riseto the figure on the right.

Reading down a vertical line we get a sequence +, · · · ,+,−, · · · ,− which we interpretas a composable sequence of arrows comprised of identities at +, the arrow w, and thenidentities at −, all of which are weak equivalences. Note the leftmost (resp. rightmost)vertical line is comprised of a sequence of identities at the codomain (resp. domain) of thek-arrow (3.2.6). By “pinching” these sequences of identities, we obtain the starting andending points of the hammock.

Reading across a horizontal line from right (the domain) to left (the codomain), we geta sequence of objects “+” or “−”, which we interpret as a zig-zag of identities togetherwith forwards (left-pointing) and backwards (right-pointing) instances of w. With theseconventions, the squiggle (3.2.6) represents the hammock:

+ oo

��

+ //

��

+ oo

w��

+ //

��

+ oo

w��

+

��

+ oo

w

��

+ w //

��

− oo w

��

+ w //

��

− oo w

��

+

w

��

+

99

w%%

EE

w

��

−%%w

99

��

w

EE

− oo w

��

+ w //

w��

− oo w

��

+ w //

��

− oo

��

−

��

− oo − // − oo w + w // − oo −On account of our conventions for the direction of horizontal composition, the hammockdescribed here is a reflection of the k-arrow displayed on [6, p. 19] in a vertical line, with“backwards” arrows point to the right. Each column will contain at least one “w”, andevery “w” in a given column will point in the same direction. This dictates the directionof the identities in that column. We leave it to the reader to verify that the hammockscorresponding to strictly undulating squiggles are “reduced” in the sense of [6, 2.1].

3.3. The 2-categorical universal property of Adj. This section is devoted to relat-ing our 2-category Adj to the generic adjunction 2-category as first studied by Schanueland Street in [28]. Our approach will be to show that the 2-category Adj established byproposition 3.2.2 enjoys the universal property they used to characterise their 2-category.

24 RIEHL AND VERITY

It follows then that these 2-categories must be isomorphic. This observation provides uswith a alternative description of Adj in terms of the structure of �, which we expoundupon in remark 3.3.8.

3.3.1. Observation (2-categories as simplicial categories). When we regard a 2-category Kas a simplicial category then• its 1-cells and 2-cells respectively define the 0-arrows and 1-arrows,• if φ0, φ1, and φ2 are three 1-arrows (2-cells) in some hom-space K(A,B) then thereexists a unique 2-arrow φ in K(A,B) with φ · δi = φi for i = 0, 1, 2 if and only ifφ0 · φ2 = φ1 in the category K(A,B), and• the rest of the structure of each hom-space K(A,B) is completely determined by thefact that it is 2-coskeletal, as is the nerve of any category.

In the terminology of example 2.2.5 the last of these observations tells is that K is a2-coskeletal simplicial category. It follows by adjunction that every simplicial functorF : sk2 L → K admits a unique extension to a simplicial functor F : L → K.3.3.2. Observation (the adjunction in Adj). In example 3.1.11, we noted that the lowdimensional data encoded in the simplicial category Adj was reminiscent of that associatedwith an adjunction. The reason that we did not commit ourselves fully to that point ofview there was that at that stage we did not actually know that Adj was a 2-category.Proposition 3.2.2 allows us to cross this Rubicon and observe that the 2-category Adj doesindeed contain a genuine adjunction:

−u

22⊥ +

f

rr η : id+ ⇒ uf ε : fu⇒ id− (3.3.3)

It turns out that this is the generic or universal adjunction, in the sense made precise inthe following proposition:

3.3.4. Proposition (a 2-categorical universal property of Adj). Suppose that K is a 2-category containing an adjunction

Au

22⊥ B

frr η : idB ⇒ uf ε : fu⇒ idA .

Then there exists a unique 2-functor Adj → K which carries the adjunction depictedin (3.3.3) to the specified adjunction in K.Proof. We know that Adj is a simplicial computad and that its non-degenerate and atomic0-arrows and 1-arrow are u, f , ε, and η. Of course sk−1 Adj is the discrete simplicialcategory with objects − and +, so we can define a simplicial functor F : sk−1 Adj → Ksimply by setting F (−) := A and F (+) := B. Now we can apply proposition 2.2.6 toextend this to a simplicial functor F : sk0 Adj → K which is uniquely determined by theequalities F (u) = u and F (f) = f and then extend that, in turn, to a simplicial functorF : sk1 Adj → K which is uniquely determined by the further equalities F (ε) = ε andF (η) = η.


Applying proposition 2.2.6 one more time, we see that extensions of the simplicial functorwe’ve defined thus far to a simplicial functor F : sk2 Adj→ K are uniquely and completelydetermined by specifying how it should act on the families of 2-arrows α(n) and β(n) intro-duced in example 3.1.11. As discussed in observation 3.3.1, we know that there exists aunique 2-arrow F (α(n)) in the 2-categoryK which satisfies the boundary conditions requiredby proposition 2.2.6 if and only if the 2-cell equation F (α(n) · δ0) ·F (α(n) · δ2) = F (α(n) · δ1)holds in K. We may compute the 2-cells that occur in these equations using the equalitieslisted in (3.1.12) and the simplicial functoriality of F on sk1 Adj to give

F (α(2r) · δ2) = ηru F (α(2r) · δ1) = u F (α(2r) · δ0) = uεr

F (α(2r+1) · δ2) = ηr+1 F (α(2r+1) · δ1) = η F (α(2r+1) · δ0) = uεrf

and so those conditions reduce to:uεr · ηru = u uεrf · ηr+1 = η

The following middle four calculationuεrf · ηr+1 = uεrf · ηruf · η = (uεr · ηru)f · η

reveals that the second of these equations follows from the first. Furthermore, the middlefour computation

uεr+1 · ηr+1u = uεr · (uf)ruε · ηrufu · ηu = uεr · ηru · uε · ηushows that we can reduce the (r + 1)th instance of the first equation to a combination ofits rth instance and the triangle identity uε · ηu = u. Consequently it follows, inductively,that all of these equations follow from that one triangle identity. The dual argument showsthat the equalities that arise from the family β(n) all reduce to the other triangle identityεf · fη = f .

We have shown that there exists a unique simplicial functor F : sk2 Adj → K whichcarries the canonical adjunction in Adj to the specified adjunction. Observation 3.3.1allows us to extend uniquely to a simplicial functor F : Adj → K. The desired universalproperty is established because a 2-functor of 2-categories is no more nor less than asimplicial functor between the corresponding simplicial categories. �

3.3.5. Corollary. The simplicial category Adj is isomorphic to the Schanuel and Street2-category of [28].

Proof. Proposition 3.3.4 tells us that our 2-category Adj satisfies the same universal prop-erty that Schanuel and Street used to characterise their 2-category. �

In [28] Schanuel and Street build their 2-category Adj directly from �+ and they ap-peal to Lawvere’s characterisation of �+ as the free strict monoidal category containinga monoid [17] in order to establish its universal property. While those authors were notthe first to discuss the existence of a 2-category whose structure encapsulates the algebraicproperties of adjunctions, their paper was the first to provide a explicit and computation-ally convenient presentation of this structure. We review their construction of Adj here asit will be useful to pass between our presentation and theirs in the sequel.

26 RIEHL AND VERITY

3.3.6. Observation (adjunctions in �+). As is common practice, we shall identify each posetP with a corresponding category whose objects are the elements p of P and which possessesa unique arrow p→ q if and only if p ≤ q in P . Under this identification, order preservingmaps are identified with functors and two order preserving maps f, g : P → Q are relatedby a unique 2-cell f ⇒ g if and only if f ≤ g under the pointwise ordering. In particular,we may regard �+ as being a full sub-2-category of Cat under the pointwise ordering ofsimplicial operators.

It is easily demonstrated that a simplicial operator α : [n] → [m] admits a left adjointαl a α (respectively right adjoint α a αr) in the 2-category �+ if and only if it carries thetop element n (respectively the bottom element 0) of [n] to the top element m (respectivelythe bottom element 0) of [m]. In particular, between the ordinals [n− 1] and [n] we havethe following sequence of adjunctions

δnn a σn−1n−1 a δn−1

n a σn−2n−1 a ... a σ1

n−1 a δ1n a σ0

n−1 a δ0n (3.3.7)

of elementary operators. We shall use the notation �∞ (respectively �−∞) to denote thesub-category of � consisting of those simplicial operators which preserve the top (respec-tively bottom elements) in each ordinal.

Of course, each identity operator stands as its own left and right adjoint. Furthermore,since adjunctions compose, we know that if α : [n]→ [m] and β : [m]→ [r] both admit left(respectively right) adjoints then so does their composite and (β◦α)l = αl◦βl (respectively(β ◦ α)r = αr ◦ βr). It follows that the act of taking adjunctions provides us with a pair ofmutually inverse contravariant functors

�op∞

(−)l

11 �−∞(−)r

qq

whose action on elementary operators may be read off from (3.3.7) as

(δin)l = σin−1 when 0 ≤ i < n (δin)r = σi−1n−1 when 0 < i ≤ n

(σin−1)l = δi+1n when 0 ≤ i ≤ n− 1 (σin−1)r = δin when 0 ≤ i ≤ n− 1.

Notice here that these formulae cover all cases because an elementary face operator δin isin �∞ if and only if i < n and is in �−∞ if and only if 0 < i whereas every elementarydegeneracy operator is in both of these subcategories.

3.3.8. Remark (the Schanuel and Street 2-category Adj). Schanuel and Street define Adjto be a 2-category with two objects, which we shall again call − and +, and with hom-categories given by

Adj(+,+) := �+, Adj(−,−) := �op+ ,

Adj(−,+) := �∞ ∼= �op−∞, and Adj(+,−) := �−∞ ∼= �op

∞


or more evocatively depicted as

−�∞∼=�op

−∞

33�op+ 77 +

�−∞∼=�op∞

ss �+

ww

The isomorphisms �∞ ∼= �op−∞ are those of observation 3.3.6.

It should come as no surprise that �+ features as the endo-hom-category on + in thisstructure. In essence this fact follows directly from Lawvere’s result, since the monad gener-ated by an adjunction is a monoid in a category of endofunctors under the strict monoidalstructure given by composition. Schanuel and Street define the composition operationsthat hold between the hom-categories of Adj in terms of the ordinal sum bifunctor

�+ × �+−⊕−

// �+

[n], [m]

α,β��

� //

7→

[n+m+ 1]

α⊕β��

[n′], [m′] � // [n′ +m′ + 1]

α⊕ β(i) :=

{α(i) i ≤ n

β(i− n− 1) + n′ + 1 i > n

which defines the strict monoidal structure on �+.Ordinal sum, regarded as a bifunctor on �+ and on its dual �op

+ , provide the compositions

Adj(+,+)× Adj(+,+)◦ // Adj(+,+) Adj(−,−)× Adj(−,−)

◦ // Adj(−,−)

on endo-hom-categories. Ordinal sum restricts to the subcategories �∞ and �−∞ to givebifunctors

�+ × �∞⊕

// �∞ �−∞ × �+⊕

// �−∞which provide the composition operations

Adj(+,+)× Adj(−,+)◦ // Adj(−,+) Adj(+,−)× Adj(+,+)

◦ // Adj(+,−)

respectively. Furthermore, the isomorphic presentations of Adj(+,−) and Adj(−,+) interms of �op

∞ and �op−∞ ensure that these restricted ordinal sum bifunctors on the duals �op

+ ,�op∞ and �op

−∞ may also be used to provide composition operations

Adj(−,−)× Adj(+,−)◦ // Adj(+,−) Adj(−,+)× Adj(−,−)

◦ // Adj(−,+)

respectively. We shall simply write

�∞ × �op+

⊕// �∞ �op

+ × �−∞⊕

// �−∞

to denote these transformed instances of the join operation, since the order of the factorsin the domain along with the dual that occurs there will disambiguate our usage.

Finally, observe that the following restriction of the ordinal sum bifunctor

�−∞ × �∞⊕

// �+

28 RIEHL AND VERITY

carries a pair of simplicial operators to a simplicial operator which preserves both top andbottom elements. So it follows that this bifunctor factors through the interval representa-tion ir : �op

+ → �+ to give a bifunctor

�−∞ × �∞⊕

// �op+

which we use to provide the last two composition actions:

Adj(−,+)× Adj(+,−)◦ // Adj(+,+) Adj(+,−)× Adj(−,+)

◦ // Adj(−,−)

A copy of the object [0] resides in each of the hom-categories Adj(−,+) and Adj(+,−)and that these correspond to the 0-arrows u and f respectively in our presentation of Adj.Furthermore the copies of the face operator δ0 : [−1] → [0] that reside in Adj(+,+) andAdj(−,−) correspond to our unit η and counit ε respectively.

4. Adjunction data

Recall an adjunction of quasi-categories is an adjunction in qCat2. The basic theoryof adjunctions is developed in section I.4. In this section, we filter the free homotopycoherent adjunction Adj by a sequence of “parental” subcomputads and use this filtra-tion to prove that any adjunction of quasi-categories—or, more precisely, any diagram in-dexed by a parental subcomputad—extends to a homotopy coherent adjunction, a diagramAdj→ qCat∞. Our proof is essentially constructive, enumerating the choices necessary tomake each stage of the extension. We conclude by proving that the appropriate spaces ofextensions, defined here, are contractible Kan complexes, the usual form of a “homotopicaluniqueness” statement in the quasi-categorical context.

In §4.1, we introduce the notion of fillable arrow, which will be used in §4.2 to defineparental subcomputads. Our aim in this section is to prove proposition 4.2.15, which showsthat any nested pair of parental subcomputads may be filtered as a countable sequenceof such, where each subcomputad is generated relative to the previous one by a finite setof fillable arrows. In §4.3, we apply this result to prove that any adjunction extends to ahomotopy coherent adjunction. In §4.4, we give precise characterisations of the homotopicaluniqueness of such extensions.

4.1. Fillable arrows.

4.1.1. Definition. An arrow a of Adj is (left) fillable if and only if• it is non-degenerate and atomic,• its codomain a0 = −, and• ai 6= a1 for all i > 1.

Write Atomn ⊂ Adjn for the subset of all atomic and non-degenerate n-arrows and writeFilln ⊂ Atomn for the subset of fillable n-arrows.

Our proof inductively specifies the data in the image of a homotopy coherent adjunctionby choosing fillers for horns corresponding to fillable arrows. We will see that the unique


fillable 0-arrow f = (−,+) behaves somewhat differently; nonetheless it is linguisticallyconvenient to include it among the fillable arrows.

4.1.2. Observation (fillable arrows and distinguished faces). Any fillable n-arrow a withn > 0 has width greater than or equal to 2 and a distinguished codimension-1 face whoseindex is k(a) := a1. Note here that 1 ≤ k ≤ n so this distinguished face may have any indexexcept for 0. On account of the graphical calculus, in which a fillable arrow a correspondsto a squiggle descending from “−” on the left to make its first turn at level k(a), we referto k(a) as the depth of a.

The fillability of a implies that no reduction steps are then required in the process offorming the distinguished face a · δk(a). Consequently, this distinguished face is again non-degenerate and has the same width as a. To further analyse this distinguished face, weneed to consider two cases:• case k(a) < n: a·δk(a) is also atomic. However, it is not fillable because non-degeneracyof a requires that there is some i > 1 such that ai = a1 +1, whence the entries of a ·δk(a)

at indices 1 and i are both equal to k(a).• case k(a) = n: a · δk(a) decomposes as fb where b is non-degenerate and atomic, haswidth one less than that of a, and has b0 = +.

We shall use the notation a� to denote the non-degenerate, atomic, and non-fillable (n−1)-arrow given by:

a� :=

{a · δk(a) when k(a) < n, andb when k(a) = n and a · δk(a) = fb.

4.1.3. Lemma. Let a be a non-degenerate and atomic n-arrow of Adj with a0 = −. Theneither it is:• a fillable arrow, or• the codimension-1 face of exactly two fillable (n+ 1)-arrows of the same width.

In the second case, both fillable (n + 1)-arrows have a as the ath1 face. One of these fil-

lable arrows, which we shall denote by a†, has depth a1 and the other has depth a1 + 1.Consequently, a† is the unique fillable (n+ 1)-arrow with the property that (a†)� = a.

Proof. If a is not fillable, then aj = a1 for some j > 1. Any arrow b admitting a as acodimension-1 face is obtained by inserting an extra line. If the arrow b is to be fillableand of the same width as a, then this line must be inserted in the kth level and used toseparate a1 from the other aj. There are exactly two ways to do this, as illustrated below:

a :=

4

3

2

1

−

+

a† :=

5

4

3

2

1

−

+

or

5

4

3

2

1

−

+

30 RIEHL AND VERITY

�

4.1.4. Lemma. Let a be a non-degenerate and atomic n-arrow of Adj with a0 = +. Thenthe composite arrow fa is a codimension-1 face of exactly one fillable (n+1)-arrow a†. The(n + 1)-arrow a† has width one greater than that of a, a†1 = n + 1, and fa = a† · δn+1. Inother words, a† is the unique fillable (n+ 1)-arrow with the property that (a†)� = a.

Proof. The construction of a† from a is illustrated in the following sequence of diagrams:

a :=

4

3

2

1

−

+

fa :=

4

3

2

1

−

+

a† :=

5

4

3

2

1

−

+

That is, we “add” an extra loop to the left and then “insert” an extra line into the bottommost space. The uniqueness of this (n+ 1)-arrow is clear. �

4.2. Parental subcomputads.

4.2.1. Definition (fillable parents). When a is a non-degenerate and atomic n-arrow inAdj which is not fillable then we define its fillable parent a† to be the fillable (n+ 1)-arrowintroduced in lemma 4.1.3 in the case where a0 = − and in lemma 4.1.4 in the case wherea0 = +. These lemmas tell us that a† is the unique fillable (n+ 1)-arrow with the propertythat (a†)� = a. Consequently, the fillable parent relation provides us with a canonicalbijection between the set Filln+1 of all fillable (n + 1)-arrows and the set Atomn \ Filln ofall non-degenerate and atomic n-arrows which are not fillable.

4.2.2. Definition (parental subcomputads of Adj). We say that a subcomputad A of Adjis parental if it contains at least one non-identity arrow and satisfies the condition that• if a is a non-degenerate and atomic arrow in A then either it is fillable or its fillableparent a† is also in A.

These conditions imply that any parental subcomputad must contain a fillable arrow. Byobservation 3.1.9, the 0th vertex of any fillable arrow may be decomposed as a compositefb. Hence, any parental subcomputad contains the 0-arrow f .

4.2.3. Example. The 0-arrow f is fillable and the subcomputad {f} ⊂ Adj that it gener-ates, as described in definition 2.2.4, has f as its only non-degenerate and atomic arrow,so is trivially a parental subcomputad. Since every parental subcomputad must contain f ,this is the minimal such.

The counit 1-arrow ε is fillable and the generated subcomputad {ε} ⊂ Adj has f , u, andε as its non-degenerate and atomic arrows. Now f is fillable and ε is the fillable parent ofu, so {ε} is a parental subcomputad.


The triangle identity 2-arrow β is fillable and the generated subcomputad {β} ⊂ Adjhas f , u, ε, η, and β as its non-degenerate and atomic arrows. Now f is fillable, ε is thefillable parent of u, and β is the fillable parent of η, so {β} is a parental subcomputad.

4.2.4. Example (a non-example). The subcomputad {α, β} ⊂ Adj generated by the tri-angle identity 2-arrows has f , u, ε, η, β, and α as its non-degenerate and atomic arrows.This is not parental, as witnessed by the fact that the 3-arrow ω of example 3.1.11 is thefillable parent of the 2-arrow α but it is not an arrow in {α, β}.4.2.5. Example. Example 3.1.11 names the 3-arrows ω and τ and the 2-arrow µ whichfeatured in the discussion of adjunction data in section 1.1. Observe that the arrows ωand τ are both fillable and that the subcomputad {ω, τ} ⊂ Adj which they generate hasf , u, ε, η, β, α, τ , ω, and µ as its non-degenerate and atomic arrows. Since τ is the fillableparent of µ, {ω, τ} is also a parental subcomputad.

Examples 4.2.3 and 4.2.5 establish a chain of parental subcomputad inclusions

{f} ⊂ {ε} ⊂ {β} ⊂ {ω, τ} ⊂ Adj.Our aim in the remainder of this section is to filter a general parental subcomputad inclu-sion A ⊂ A′ as a countable tower of parental subcomputads, with each sequential inclusionpresented as the pushout of an explicit map. To describe each “attaching step,” we turnour attention to certain families of simplicial categories. Recall the simplicial categories2[X] introduced in 2.1.3.

4.2.6. Notation. Let 3[X] denote the simplicial category with objects 0, 1, and 2, non-trivial hom-sets 3[X](0, 1) := X, 3[X](1, 2) := ∆0, 3[X](0, 2) := X ? ∆0, and whose onlynon-trivial composition operation is defined by the canonical inclusion:

3[X](1, 2)× 3[X](0, 1) = ∆0 ×X ∼= X ↪→ X ?∆0 = 3[X](0, 2)

Here we define 3[X](2, 1) = 3[X](1, 0) = 3[X](2, 0) = ∅ and 3[X](0, 0) = 3[X](1, 1) =3[X](2, 2) = ∗. A simplicial functor F : 3[X]→ K is determined by the following data:• a 0-arrow f : B → A and an object C of K and• a pair of simplicial maps g : X → K(C,B) and h : X ? ∆0 → K(C,A) such that thefollowing square commutes:

X //

g

��

X ?∆0

h��

K(C,B)K(C,f)

// K(C,A)

(4.2.7)

The map h : X ? ∆0 → K(C,A) may be described in terms of Joyal’s slicing constructionof definition I.2.4.2, by giving a 0-arrow a : C → A (the image of the ∆0) and a simplicialmap h : X → K(C,A)/a. The commutative square (4.2.7) transposes to the commutative

32 RIEHL AND VERITY

square:

Xh //

g

��

K(C,A)/a

π

��

K(C,B)K(C,f)

// K(C,A)

(4.2.8)

This data may be captured by a single map l : X → K(C, f)/a whose codomain is thepullback of K(C,A)/a along K(C, f), i.e., the slice K(C, f)/a of the map K(C, f) over a asdefined in remark I.2.4.14. Thus, a simplicial functor F : 3[X] → K is determined by thefollowing data:• a pair of 0-arrows f : B → A and a : C → A of K and• a simplicial map l : X → K(C, f)/a.

4.2.9. Definition. A fillable n-arrow a gives rise to a corresponding simplicial functor Fainto Adj defined as follows:• If a1 < n define Fa : 2[∆n]→ Adj to be the simplicial functor induced out of the genericn-arrow 2[∆n] by a; cf. 2.1.3.• If a1 = n define Fa : 3[∆n−1]→ Adj so that it:◦ maps the objects 1 to +, 2 to −, and 0 to aw(a), the domain of a,◦ maps the hom-set 3[∆n−1](1, 2) ∼= ∆0 to Adj(+,−) by the unique simplicial mapwhich corresponds to the 0-arrow f ,◦ maps the hom-set 3[∆n−1](0, 1) ∼= ∆n−1 to Adj(aw(a),+) by the unique simplicialmap which corresponds to the (n− 1)-arrow a�, and◦ maps the hom-set 3[∆n−1](0, 2) ∼= ∆n to Adj(aw(a),−) by the unique simplicial mapwhich corresponds to the n-arrow a itself.

The relation a ·δn = fa� implies that these actions are compatible with the compositionstructures of 3[∆n−1] and Adj.

4.2.10. Lemma (extending parental subcomputads). Suppose that A is a parental sub-computad of Adj and that a is a fillable n-arrow of depth k := a1 which is not a memberof A, and let A′ be the subcomputad of Adj generated by A ∪ {a}. Suppose also that thecodimension-1 face a · δi is a member of A for each i ∈ [n] with i 6= k. Then A′ is also aparental subcomputad, and we may restrict the the functor Fa of definition 4.2.9 to expressthe inclusion A ↪→ A′ as a pushout

2[Λn,k] ��

//

Fa

��

2[∆n]

Fa

��

A ��

// A′

(4.2.11)


when k < n and as a pushout

3[∂∆n−1] ��

//

Fa

��

3[∆n−1]

Fa

��

A ��

// A′

(4.2.12)

when k = n.

Proof. Because a is the fillable parent of a�, this non-degenerate, atomic, and non-fillable(n− 1)-arrow cannot be an element of the parental subcomputad A. Now by assumptionall of the faces a · δi with i 6= k are contained in A, so a and a� are the only two atomicarrows which are in A′ but are not in A. The first of these is fillable and the second hasthe first as its fillable parent; hence, A′ is again parental.

To verify the that the squares given in the statement are pushouts of simplicial categories,observe that extensions of F : A → K to a simplicial functor F ′ : A′ → K are completelydetermined by specifying what the atomic arrows a and a� should be mapped to in K,subject to domain, codomain, and face conditions imposed by the simplicial functor F .Specifically, to make this extension we must provide:• case k < n: an n-arrow g in K with the property that g · δi = F (a · δi) for all i 6= k,i.e., a simplicial functor g : 2[∆n]→ K which makes the following square commute:

2[Λn,k] ��

//

g

��

2[∆n]

g

��

AF

// K

• case k = n: an (n − 1)-arrow g and an n-arrow h in K with the property that andg · δi = F (a� · δi) and h · δi = F (a · δi) for all i 6= n, and also that h · δn = (Ff)g. Inother words, by 4.2.6, we require a simplicial functor h : 3[∆n−1]→ K which makes thefollowing square commute:

3[∂∆n−1] ��

//

h

��

3[∆n−1]

h

��

AF

// K

�

By iterating lemma 4.2.10, we have proven:

4.2.13. Corollary. Suppose that A is a parental subcomputad of Adj and that X is a set offillable arrows, each of which is not in A but has the property that every face except the oneindexed by its depth is in A. Let A′ be the subcomputad of Adj generated by A∪X. Then

34 RIEHL AND VERITY

A′ is a parental subcomputad and the inclusion A ↪→ A′ may be expressed as a pushout

(∐a∈X

a1<dim(a)

2[Λdim(a),a1 ]) t (∐a∈X

a1=dim(a)

3[∂∆dim(a)−1]) ��

//

〈Fa〉a∈X��

(∐a∈X

a1<dim(a)

2[∆dim(a)]) t (∐a∈X

a1=dim(a)

3[∆dim(a)−1])

〈Fa〉a∈X��

A ��

// A′

(4.2.14)of simplicial categories.

4.2.15. Proposition. Suppose that A and A′ are parental subcomputads of Adj and thatA ⊆ A′. Then we may filter this inclusion as a countable tower of parental subcomputadsA0 ⊆ A1 ⊆ A2 ⊆ · · · (A = A0 and A′ =

⋃i≥0 Ai) in such a way that for each i ≥ 1 there

is a non-empty and finite set Xi of arrows such that(i) each arrow in Xi is fillable, is not contained in Ai−1, but has the property that

every face except the one indexed by its depth is in Ai−1, and(ii) the subcomputad Ai is generated by Ai−1 ∪Xi.

Hence, the inclusion map A ↪→ A′ may be expressed as a countable composite of inclusionsall of which may be constructed as pushouts of the form (4.2.14).

Proof. Let X denote the set of all fillable arrows which are in A′ and are not in A, and letXw,k,n denote the subset of those arrows which have width w, depth k, and dimension n.Now any non-degenerate arrow of Adj must have dimension which is strictly less than itswidth, and it is clear there can only be a finite number of non-degenerate arrows of anygiven width. The depth of any fillable arrow is always less than or equal to its dimension,so it follows that Xw,k,n is always finite and that it is empty unless k ≤ n < w.

Now order those index triples (w, k, n) which have k ≤ n < w under the lexicographicordering: for i ≥ 1, let (wi, ki, ni) index the subsequence of that linear ordering of thoseindex triples for which Xw,k,n is non-empty, and write Xi := Xwi,ki,ni . Let Ai be thesubcomputad of Adj generated by A ∪ (

⋃ij=1Xj) and observe that this family filters the

inclusion A ⊆ A′ since, by construction, the union of the subcomputads Ai is A′, thesubcomputad generated by A ∪ (

⋃i≥1Xi).

We complete our proof by induction on the index i, starting from the parental subcom-putad A0 = A. Adopt the inductive hypothesis that for all indices j < i the subcomputadAj is parental and condition (i) holds. For the inductive step, it suffices to check that allof the arrows in Xi satisfy condition (i) with respect to Ai−1; this amounts to verifyingthat the appropriate codimension-1 faces are in Ai−1. Applying corollary 4.2.10 to the setXi and the subcomputad Ai−1, which is parental by the inductive hypothesis, it followsthat Ai is again parental.

Observe that Ai−1 is the smallest subcomputad of Adj which contains A and all fillablesimplices in A′ which have• width less than wi, or


• width wi and depth less than ki, or• width wi, depth ki, and dimension less than ni.

Consider an arrow a in Xi, which has width wi, depth ki, and dimension ni. To completethe inductive step, it remains only to show that the face a · δl is a member of Ai−1 forevery l ∈ [ni] which is not equal to the depth ki. The details in each case, while tedious,are entirely straightforward.• case l 6= ki − 1, l 6= ki: Under this condition the line numbered l is not one of thoseseparating the level a1 from the other entries of a. It follows that the removal of line lwill not cause entry a1 to be eliminated by a reduction step, to become − or +, or toend up in the same space as another later entry. In other words, a · dl is fillable if it isnon-degenerate and atomic with depth ki − 1 if l < k1 − 1 and depth ki if l > ki.

Since Adj is a simplicial computad, a · δl may be expressed uniquely as a composite(b1 · α1) ◦ (b2 · α2) ◦ ... ◦ (br · αr) in which each bj is non-degenerate and atomic arrow ofA′ and each αj is a degeneracy operator. For the reasons just observed, b1 is a fillablearrow of width less than or equal to wi, depth less than or equal to ki, and dimensionstrictly less than ni. Hence, b1 is a member of the subcomputad Ai−1.

Furthermore, b1 has width greater or equal to 2 (because b1 6= − and b1 6= +) sothe width of each bj with j > 1 is less than or equal to wi − 2. Consequently, whenj > 1, then bj is either a fillable arrow of width less than or equal to wi − 2 or it has afillable parent of width less than or equal to wi − 1. In either case, bj is a member ofthe subcomputad Ai−1. As a · δl is a composite of degenerate images of arrows whichare all in Ai−1, it too lies in Ai−1.• case l = ki − 1: As observed in 4.2.2, the parental subcomputad Ai−1 contains thefillable 0-arrow f . Since a is not in Ai−1, we know that it must have width greater thanor equal to 2 and dimension greater than or equal to 1. The only fillable arrow of width2 is ε = (−, 1,−), for which the depth ki = 1, l = ki− 1 = 0, and we have ε · δ0 = − anidentity, which is certainly in Ai−1. So from hereon we may assume that wi > 2. Withthis assumption, a1 > a2 6= − and thus ki ≥ 2, from which it follows that on removingline l = ki−1 the resulting face a · δl must again be atomic. Now we have two subcases:◦ case a1 = a2 + 1: The line l = ki − 1 = a1 − 1 separates the levels of a1 and a2, sowhen we remove it to form the face a ·δl we must also perform at least one reductionstep. This implies that a · δl has width is less than or equal to wi − 2 and that thisface is possibly degenerate. There is a unique atomic and non-degenerate arrow band a unique degeneracy operator α such that a · δl = b · α. The arrow b is eithera fillable arrow of width less than or equal to wi − 2 or it has a fillable parent ofwidth less than or equal to wi − 1. In either case, it follows that b, and thus itsdegenerated partner a · δl, is a member of Ai−1.◦ case a1 > a2 + 1: The line l separates the level a1 from the level immediately aboveit, which contains neither a0 nor a2. So when we remove that line to form the facea · δl no reduction steps are required and this face must again be non-degenerate.However, since the arrow a is non-degenerate there must be some j > 2 such thataj = a1− 1 and hence a · δl is not fillable. Now lemma 4.1.3 implies that the fillable

36 RIEHL AND VERITY

parent (a ·δl)† has depth a1−1, which is one less than the depth ki = a1 of a. As a ·δlis a member of the parental subcomputad A′, its fillable parent is also in A′, andwe may apply by the characterisation of Ai−1 to conclude that this fillable parent(a · δl)†, and thus its face a · δl, is in the parental subcomputad Ai−1. �

4.3. Homotopy Coherent Adjunctions. In this section, we use proposition 4.2.15 toshow that every adjunction of quasi-categories gives rise to a simplicial functor Adj →qCat∞ which carries the canonical adjunction in Adj to the chosen adjunction of quasi-categories.

4.3.1. Recall (2-categories from quasi-categorically enriched categories). Recall, from ob-servation I.3.1.2, that the homotopy category construction h : qCat → Cat gives rise to afunctor h∗ : qCat-Cat → 2-Cat which reflects the category of quasi-categorically enrichedcategories qCat-Cat into its full sub-category of 2-categories 2-Cat. The 2-category h∗Kis constructed by applying h to each of the hom-spaces of K. We write K2 := h∗K for the2-category associated to a quasi-categorically enriched category K. When F : K → L is asimplicial functor of quasi-categorically enriched categories, we write F2 := h∗F : K2 → L2

for the associated 2-functor. We shall also adopt the notation QK : K → K2 for the manifestquotient simplicial functor, the component at K of the unit of the reflection h∗.

Given this relationship, we shall use the 2-cell notation φ : f ⇒ g to denote a 1-arrow with0-arrow faces f = φ · δ1 and g = φ · δ0. This notation is consistent with the correspondingusage in the 2-category K2, since φ is a representative of a genuine 2-cell φ : f ⇒ g in there.

4.3.2. Observation (adjunctions in a quasi-categorically enriched category). Suppose that Kis a quasi-categorically enriched category. An adjunction f a u : A→ B in the 2-categoryK2 may be presented by the following information in K itself:• a pair of 0-arrows u ∈ K(A,B) and f ∈ K(B,A),• a pair of 1-arrows η ∈ K(B,B) and ε ∈ K(A,A) which represent the unit and counit2-cells in K2 and whose boundaries are depicted in the following pictures

idBη +3 uf fu

ε +3 idA ,

and• a pair of 2-arrows α ∈ K(A,B) and β ∈ K(B,A) which witness the triangle identitiesand whose boundaries are depicted in the following pictures:

ufuuε

�$

fufεf

�%u

u·σ0+3

ηu :B

u ff ·σ0

+3

fη 9A

fα β

This information is not uniquely determined by our adjunction since it involves choices ofrepresentative 1-arrows for its unit and counit 2-cells and choices of witnessing 2-arrowsfor its triangle identities.

This data used to present an adjunction in K uniquely determines a simplicial functorT : {α, β} → K whose domain is the subcomputad of Adj generated by the triangle identity


2-arrows, as in example 4.2.4, and whose action on non-degenerate and atomic arrows isgiven by T (f) = f , T (u) = u, T (ε) = ε, T (η) = η, T (β) = β, and T (α) = α.

Since adjunctions are defined equationally in a 2-category, they are preserved by any 2-functor. It follows, therefore, that adjunctions are preserved by the 2-functor F2 : K2 → L2

associated with any simplicial functor F : K → L of quasi-categorically enriched categories.Explicitly, the adjunction displayed above transports along F to give an adjunction F (f) aF (u) in L which is presented by unit and counit 1-arrows F (η) and F (ε) and 2-arrows F (α)and F (β) which witness its triangle identities.

4.3.3. Notation. For the remainder of this section we shall assume that K and L arequasi-categorically enriched categories. Furthermore, we shall assume that we have beengiven a simplicial functor P : K� L which is a local isofibration in the sense that its actionP : K(A,B) � L(PA, PB) on each hom-space is an isofibration of quasi-categories. Wewill also fix an adjunction

Au

22⊥ B

frr

in K2 that is presented in K by unit and counit 1-arrows η : idB ⇒ uf and ε : fu⇒ idA and2-arrows α and β which witness its triangle identities as in observation 4.3.2. To remind thereader of our standing hypotheses, we might write “suppose K has an adjunction (f a u, ε),”listing in parentheses the data in K chosen to present an adjunction in K2.

4.3.4. Observation (the internal universal property of the counit). Suppose that C is anarbitrary object of K. The representable simplicial functor K(C,−) : K → qCat∞ carriesour adjunction f a u in K to an adjunction

K(C,A)K(C,u)

22⊥ K(C,B)

K(C,f)rr

of quasi-categories with unit K(C, η) and counit K(C, ε). On applying proposition I.4.4.8 tothis adjunction of quasi-categories, we find that if a is a 0-arrow in K(C,A) then εa : fua⇒a may be regarded as being an object of the slice quasi-category K(C, f)/a wherein it is aterminal object. So, in particular, it follows that if ∂∆n−1 → K(C, f)/a is a simplicial mapwhich carries the vertex {n− 1} of ∂∆n−1 to the object εa then it may be extended alongthe inclusion ∂∆n−1 ↪→ ∆n−1 to a simplicial map ∆n−1 → K(C, f)/a.

On consulting 4.2.6, we discover that simplicial maps ∂∆n−1 → K(C, f)/a (respectively∆n−1 → K(C, f)/a) which carry {n− 1} to εa stand in bijective correspondence to simpli-cial functors 3[∂∆n−1] → K (respectively 3[∆n−1] → K) which carry the 0-arrow {0} of3[∂∆n−1](1, 2) = ∆0 to f , the 0-arrow {n− 1} of 3[∂∆n−1](0, 1) to ua, and the 1-arrow{n− 1, n} of 3[∂∆n−1](0, 2) = ∆n−1 ? ∆0 ∼= ∆n to εa. It follows that the universal prop-erty of the counit 1-arrow ε discussed above simply posits the existence of the lift T ′ in the

38 RIEHL AND VERITY

following diagram3[∂∆n−1]

� _

��

T // K

3[∆n−1]

T ′

;;

so long as T ({0} : 1→ 2) = f , T ({n− 1} : 0→ 1) = ua, and T ({n− 1, n} : 0→ 2) = εa.

To prove a relative version of this result, we require the following lemma:

4.3.5. Lemma (a relative universal property of terminal objects). Suppose that E and Bare quasi-categories which possess terminal objects and that p : E � B is an isofibrationwhich preserves terminal objects, in the sense that if t is is terminal in E then pt is terminalin B. Then any lifting problem

∂∆n u //� _

��

E

p��

∆nv// B

with n > 0 has a solution so long as u carries the vertex {n} to a terminal object in E.

Proof. Using the universal property of the terminal object t := u{n} in E we may extendthe map u : ∂∆n → E to a map w : ∆n → E. Now we have two maps pw, v : ∆n → B bothof which restrict to the boundary ∂∆n to give the same map pu : ∂∆n → B. From thesewe can construct a map h : ∂∆n+1 → B with hδn+1 = pw and hδn = v by starting with thedegenerate simplex pwσn : ∆n+1 → B, restricting to its boundary, and then replacing thenth face in this sphere with v : ∆n → B. Of course h maps the object {n+ 1} to the objectpt which is terminal in B, so it follows that we may extend it to a map k : ∆n+1 → B.We may also construct a map g : Λn+1,n → E by restriction from the degenerate simplexwσn : ∆n+1 → E and observe that we may decompose the commutative square of thestatement into the following composite of commutative squares:

∂∆n � � δn//

� _

��

Λn+1,n g//

� _

��

E

p��

∆n � �

δn// ∆n+1

k//

l

;;

B

Since the central vertical of this square is an inner horn inclusion and its right hand verticalis an isofibration of quasi-categories, it follows that the lifting problem on the right has asolution l : ∆n+1 → E as marked. Now it is clear that the map lδn : ∆n → E provides asolution to the original lifting problem. �


4.3.6. Proposition (the relative internal universal property of the counit). If K has anadjunction (f a u, ε), then the following lifting problem has a solution

3[∂∆n−1]T //

� _

��

KP��

3[∆n−1]S// L

provided that T ({0} : 1 → 2) = f , T ({n− 1} : 0 → 1) = ua, and T ({n− 1, n} : 0 → 2) =εa for some 0-arrow a ∈ K(C,A).

Proof. On consulting 4.2.6, we see that we may translate the lifting problem of the state-ment into a lifting problem of the following form

∂∆n−1 t //� _

��

K(C, f)/a

P��

∆n−1s// L(PC, Pf)/Pa

in simplicial sets. The upper horizontal map t : ∂∆n−1 → K(C, f)/a carries the vertex{n− 1} to the object εa of K(C, f)/a, which is terminal in there by observation 4.3.4.Furthermore, using the local isofibration property of the simplicial functor P it is easilyverified that the vertical map on the right of this square is an isofibration of quasi-categories.This map carries the terminal object εa of K(C, f)/a to the object P (εa) = (Pε)(Pa) ofL(PC, Pf)/Pa, which is again terminal since Pε is the counit of the transported adjunctionPf a Pu in L. Applying lemma 4.3.5, we obtain the desired lift. �

4.3.7. Observation. As an easier observation of a similar ilk, note that the fact that oursimplicial functor P : K � L is a local isofibration implies that that we may solve thelifting problem

2[Λn,k]T //

� _

��

KP��

2[∆n]S// L

whenever n ≥ 2 and 0 < k < n.

Combining these observations with the results of §4.2, we obtain the following liftingresult:

4.3.8. Theorem. Suppose that A and A′ are parental subcomputads of Adj and that A ⊆A′. Furthermore, assume that A contains the 0-arrows u and f and the 1-arrow ε. Then

40 RIEHL AND VERITY

if K has an adjunction (f a u, ε), we may solve the lifting problem

AT //

� _

��

KP��

A′S// L

so long as T (f) = f , T (u) = u, and T (ε) = ε.

Proof. We know, by proposition 4.2.15, that we may filter the inclusion A ⊆ A′ as acountable sequence of inclusions all of which may be constructed as pushouts of the form(4.2.11) or (4.2.12). It follows that we may reduce this result to the case where A′ is aparental subcomputad which extends A by the addition of a single fillable n-arrow a ofdepth k := a1 as discussed in the statement of lemma 4.2.10.

Now consider the two cases identified in lemma 4.2.10. The first of of these is the easycase k < n, in which situation we have the following commutative diagram

2[Λn,k]Fa//

� _

��

AT //

� _

��

K

P

��

2[∆n]Fa// A′

S// L

where the square on the left is the pushout of (4.2.11). Now observation 4.3.7 provides uswith a solution for lifting problem which is the composite of these two squares. Then wemay use that lift and the universal property of the pushout on the left to construct thesolution we seek for the lifting problem on the right.

In the case where k = n our argument is a little more involved, but here again we startwith a commutative diagram

3[∂∆n−1]Fa//

� _

��

AT //

� _

��

K

P

��

3[∆n−1]Fa// A′

S// L

where the square on the left is the pushout of (4.2.12). Consulting the definition of thesimplicial functor Fa : 3[∆n−1] → Adj, as given in 4.2.9, we see that it maps the 0-arrow{0} : 1 → 2 to f and it maps the n-arrow id[n] : 0 → 2 to a, so it maps {n− 1, n} : 0 → 2to a · {n− 1, n}. To calculate this edge we delete the lines numbered 0, 1, ..., n − 2 andthen reduce. However a is a fillable n-arrow of depth k = n so it has a0 = −, it is atomicso ai 6= −,+ for 0 < i < w(a), and ai 6= a1 for all i > 1; these facts together imply thatn = a1 > ai for 1 < i < w(a). There are now two cases to consider, depending on whetherthe domain aw(a) of a is − or +. In the first of these, the removal of lines 0, 1, ..., n − 2leaves a string of the form (−, 1,−,−, ...,−) in which the sequence of trailing − symbols


is of odd length, so this reduces to ε = (−, 1,−). In the second, the removal of those linesleaves a string of the form (−, 1,−,−, ...,−,+) where again the sequence of − symbols isof odd length, so this reduces to εf = (−, 1,−,+). The second of these computations isillustrated in the following sequence of squiggle pictures:

4

3

2

1

−

+

delete 1

−

+

reduce 1

−

+

By assumption, T maps f to the 0-arrow f : B → A and it maps ε to the 1-arrow ε,so it follows that TFa({n− 1, n} : 0 → 2) is equal to ε when aw(a) is − and is equal to εfwhen aw(a) is +. In either case the map TFa : 3[∂∆n−1] → K conforms to the conditionsof proposition 4.3.6, with a = idA in the first case and a = f in the second. Applyingthat result, we may find a solution for the lifting problem expressed by this compositerectangle. Then we may use that lift and the universal property of the pushout on the leftto construct the solution we seek for the lifting problem on the right. �

Having cleared the heavy lifting, we are now in a position to prove that it is possible toextend every adjunction in K2 to a homotopy coherent adjunction in K.4.3.9. Theorem (homotopy coherence of adjunctions I). If K has an adjunction (f a u, ε),then there exists a simplicial functor H : Adj → K for which H(f) = f , H(u) = u, andH(ε) = ε.

Proof. By example 4.2.3, the subcomputad {ε} ⊂ Adj is a parental subcomputad whosenon-degenerate and atomic arrows are f , u, and ε. Consequently, there exists a simplicialfunctor T : {ε} → K which is uniquely determined by the fact that it maps those generatorsto the corresponding arrows f , u, and ε in K respectively. Now we have a lifting problem

{ε} T //� _

��

K

!

��

Adj!//

H

??

1

where 1 denotes the terminal simplicial category whose only hom-set is ∆0. Becauseeach hom-space of K is a quasi-category, the right hand vertical in this square is a localisofibration. Applying theorem 4.3.8, we obtain the dashed lift H : Adj → K which, byconstruction, has the properties asked for in the statement. �

4.3.10. Remark. Applying theorem I.6.1.4 to the characterisation of adjunctions found inexample I.5.0.4, we see that a functor f : B → A is a left adjoint if and only if the slicequasi-category f/a has a terminal object for each vertex a ∈ A. In this case, theorem I.6.1.4

42 RIEHL AND VERITY

supplies a right adjoint u and counit ε : fu⇒ idA in qCat2. On choosing any representative1-arrow for that counit, theorem 4.3.9 extends this data to a simplicial functor H : Adj→qCat∞.

4.3.11. Theorem (homotopy coherence of adjunctions II). If K has an adjunction (f au, ε, η, β), there exists a simplicial functor H : Adj → K for which H(f) = f , H(u) = u,H(ε) = ε, H(η) = η, and H(β) = β.

Proof. We follow the same pattern of argument as in the proof of theorem 4.3.9. This startsby observing that example 4.2.3 tells us that the subcomputad {β} ⊂ Adj is a parentalsubcomputad whose non-degenerate and atomic arrows are f , u, ε, η, and β. It followsthen that there exists a simplicial functor T : {β} → K which is uniquely determined bythe fact that it maps those generators to the corresponding arrows f , u, ε, η, and β in Krespectively. Applying theorem 4.3.8, we again construct a simplicial functor H : Adj→ Kwhich, by construction, satisfies the conditions of the statement. �

4.3.12. Remark. Note that theorem 4.3.11 does not impose any conditions concerning theaction of the simplicial functor H : Adj → K on the other triangle identity 2-arrow α.In general, while H(α) is a 2-arrow which witnesses the other triangle identity of f a uthere is no reason why it should be equal to the particular witness α that we fixed inobservation 4.3.2. Indeed it is possible that there may be no simplicial functorH : Adj→ Kwhich simultaneously maps both of the 2-arrows α and β to that chosen pair of witnessesfor the triangle identities.

4.3.13. Definition. We know by proposition 3.3.4 that there exists a unique 2-functorF : Adj→ K2 which carries the canonical adjunction in Adj to the chosen adjunction f a uin K2. If this 2-functor lifts through the quotient simplicial functor from K to K2 as in thefollowing diagram

Adj H //

F !!

KQK��

K2

then we say that the dashed simplicial functor H : Adj→ K is a lift of our adjunction f a uto a homotopy coherent adjunction in K. More explicitly, H is any simplicial functor whichmaps u and f to the corresponding 0-arrows of the adjunction f a u and which maps ε and ηto representatives for the unit and counit of that adjunction. As an immediate consequenceof theorem 4.3.11, every adjunction in K2 lifts to a homotopy coherent adjunction in K.4.4. Homotopical uniqueness of homotopy coherent adjunctions. We conclude thissection by proving that the space of all lifts of an adjunction to a homotopy adjunctionis not only non-empty, as guaranteed by theorem 4.3.11, but is also a contractible Kancomplex. In other words, this result says that lifts of an adjunction to a homotopy coherentadjunction are “homotopically unique”.


4.4.1. Observation (simplicial enrichment of simplicial categories). We may apply the prod-uct preserving exponentiation functor (−)X : sSet→ sSet to the hom-spaces of any simpli-cial category K to obtain a simplicial category KX . This construction defines a bifunctorsSetop × sSet-Cat → sSet-Cat, and there exist canonical natural isomorphisms K∆0 ∼= Kand KX×Y ∼= (KX)Y which obey manifest coherence conditions.

Given an action of this kind of sSet on sSet-Cat, we may construct an enrichment of thelatter to a (large) simplicial category. Specifically, we take the n-arrows between simplicialcategories K and L to be simplicial functors F : K → L∆n . The action of � on these isgiven by F · α := Lα ◦ F , and we compose F with a second such n-simplex G : L →M∆n

by forming the composite:

K F // L∆n G∆n

// (M∆n)∆n ∼= M(∆n×∆n) M∇ //M∆n

The associativity and identity rules for this composition operation are direct consequencesof the fact that for anyX the diagonal map∇ : X → X×X and the unique map ! : X → ∆0

obey the co-associativity and co-identity rules. Under this enrichment by (possibly large)simplicial sets, the construction KX becomes the simplicial cotensor of K by X.

We write icon(K,L) to denote the (possibly large) simplicial hom-space between sim-plicial categories. The notation “icon” is chosen here because a 1-simplex in icon(K,L)should be thought of as analogous to an identity component oplax natural transformationin 2-category theory, as defined by Lack [16]. In particular, the simplicial functors K → Lserving as the domain and the codomain of a 1-simplex in icon(K,L) agree on objects.

The universal property of LX as a cotensor may be expressed as a natural isomorphismicon(K,L)X ∼= icon(K,LX) and, in particular, it provides a natural bijection betweensimplicial maps X → icon(K,L) and simplicial functors K → LX .

We will be interested in fibers of maps icon(A′,K) → icon(A,K) associated to anidentity-on-objects inclusion A ↪→ A′ between small simplicial categories. It is easilychecked, using the fact that the hom-spaces of K are all small simplicial sets, that any suchfibre will be a small simplicial set.

For the remainder of this section we shall assume that K and L denote quasi-categoricallyenriched categories. The icon enrichment of sSet-Cat is homotopically well-behaved withrespect to local isofibrations and relative simplicial computads, in the precise sense for-malised in the next lemma.

4.4.2. Lemma. Suppose that P : K� L is a simplicial functor which is a local isofibration,and suppose that I : A ↪→ B is relative simplicial computad. Furthermore assume eitherthat P is surjective on objects or that I acts bijectively on objects. Then the Leibnizsimplicial map

icon(I, P ) : icon(B,K) −→ icon(A,K)×icon(A,L) icon(B,L)

is a fibration in Joyal’s model structure.

44 RIEHL AND VERITY

Proof. We wish to prove that every lifting problem

X� _

i

��

// icon(B,K)

icon(I,P )��

Y // icon(A,K)×icon(A,L) icon(B,L)

whose left-hand vertical i : X ↪→ Y is a trivial cofibration in the Joyal model structure onsSet, has a solution. This lifting problem transposes into the corresponding problem:

A� _I��

// KY

hom(i,P )��

B // KX ×LX LY

Here the simplicial functor hom(i, P ) on the right is surjective on objects whenever P is,and its action on each hom-set is the Leibniz map

hom(i, P ) : K(C,D)Y → K(C,D)X ×L(PC,PD)X L(PC, PD)Y ,

which is a trivial fibration of quasi-categories because P : K(C,D) � L(PC, PD) is anisofibration of quasi-categories and i : X ↪→ Y is a trivial cofibration in Joyal’s modelstructure.

Now by definition 2.1.4 we know that I : A ↪→ B can be expressed as a countablecomposite of pushouts of inclusions ∅ ↪→ 1 and 2[∂∆n] ↪→ 2[∆n] for n ≥ 0. FurthermoreI is bijective on objects if and only if that decomposition doesn’t contain any pushoutsof the inclusion ∅ ↪→ 1. So it follows that it is enough to check that we may solve liftingproblems of the forms:

∅� _��

// KY

hom(i,P )��

1 // KX ×LX LY

2[∂∆n]� _

��

// KY

hom(i,P )��

2[∆n] // KX ×LX LY

Now solutions to problems like those on the right are guaranteed by the fact that theactions of hom(i, P ) on hom-spaces are trivial fibrations. Furthermore, if P is surjectiveon objects then we can solve problems like those on the left. Otherwise, when I is bijectiveon objects we need not solve any such problems. �

Special cases of lemma 4.4.2 imply that icon(I,K) : icon(B,K) � icon(A,K) is anisofibration of quasi-categories if A is a simplicial computad and A ↪→ B is a relativesimplicial computad.

4.4.3. Observation. Translating the proof of lemma 4.4.2 to the marked model structureof I.2.3.8, we obtain a corresponding result for relative simplicial computads and localisofibrations of categories enriched in naturally marked quasi-categories. In particular,


if K has naturally marked hom-spaces and A is a simplicial computad, then the spaceicon(A,K) is a naturally marked quasi-category.

We will be interested in the isomorphisms in these spaces of icons.

4.4.4. Lemma. Suppose that F : K� L is a simplicial functor which is locally conservativein the sense that its action F : K(A,B)→ L(FA, FB) on each hom-space reflects isomor-phisms. Suppose also that A is a simplicial computad and that I : A ↪→ B is a relativesimplicial computad that is bijective on objects. Then the Leibniz simplicial map

icon(I, F ) : icon(B,K) −→ icon(A,K)×icon(A,L) icon(B,L)

is a conservative functor of quasi-categories.

Proof. We work in the marked model structure, where we know that a simplicial map isconservative if and only if it has the right lifting property with respect to the inclusion2 ↪→ 2] of the unmarked 1-simplex into the marked 1-simplex. Transposing this liftingproperty, we find that it is equivalent to postulating that every lifting problem

A� _

��

// K2]

hom(i,F )��

B // K2 ×L2 L2]

has a solution. SinceA ↪→ B is both a relative simplicial computad and bijective on objectsit is expressible as a composite of pushouts of inclusions of the form 2[∂∆n] ↪→ 2[∆n], andit suffices to consider the case when A ↪→ B is an inclusion of this form. This amounts toshowing that each lifting problem

∂∆n� _

��

// K(A,B)2]

hom(i,F )��

∆n // K(A,B)2 ×L(FA,FB)2 L(FA, FB)2]

has a solution, which transposes to give the following lifting problem:

(∂∆n × 2]) ∪ (∆n × 2)� _

��

// K(A,B)

F

��

∆n × 2] // L(FA, FB)

The marked simplicial sets on the left have the same underlying simplicial sets and differonly in their markings. Consequently, the (unique) existence of the solution to this latterproblem follows immediately from the assumption that F is locally conservative. �

4.4.5. Definition. The vertices of the quasi-category icon(Adj,K) are precisely the homo-topy coherent adjunctions in K, so we write cohadj(K) = icon(Adj,K) for the space ofhomotopy coherent adjunctions in K.

46 RIEHL AND VERITY

4.4.6. Lemma. The space of homotopy coherent adjunctions in K is a (possibly large) Kancomplex.

Proof. Lemma 4.4.2 implies that cohadj(K) is a quasi-category, so to demonstrate that itis a Kan complex we need only show that all of its arrows are isomorphisms. This problemreduces to a 2-categorical argument. Observe that the quotient simplicial functor QK : K →K2 is locally conservative simply because, by definition, the isomorphisms of the quasi-category K(A,B) are those arrows which map to isomorphisms in the homotopy categoryK2(A,B) = h(K(A,B)). Lemma 4.4.4 implies that the functor cohadj(QK) : cohadj(K)→cohadj(K2) is conservative. So it follows that the arrows of cohadj(K) are isomorphisms ifthis is the case for cohadj(K2).

It is easy to see that cohadj(K2) is a category. The 2-categorical universal property ofAdj established in proposition 3.3.4 furnishes a concrete description of this category:• objects are adjunctions (f a u, ε, η) in K2,• arrows (φ, ψ) : (f a u, ε, η) → (f ′ a u′, ε′, η′) consist of a pair of 2-cells φ : f ⇒ f ′ andψ : u⇒ u′ in K2 which satisfy the equations ε′ · (φψ) = ε and η′ = (ψφ) · η, and• identities and composition are given component-wise.The isomorphisms in this category are those pairs whose constituent 2-cells are invertible.

Given any arrow (φ, ψ) : (f a u, ε, η)→ (f ′ a u′, ε′, η′), we may construct the mates of the2-cells φ and ψ under the the given adjunctions, that is to say the following composites:

ψ′ := f ′f ′η +3f ′uf

f ′ψf +3f ′u′fε′f +3f φ′ := u′

ηu′ +3ufu′uφu′ +3uf ′u′

uε′ +3u

We leave it to the reader to verify that φ′ is inverse to ψ and that ψ′ is inverse to φ. �

The following proposition strengthens lemma 4.4.2 when we restrict our attention toinclusions of parental subcomputads of Adj.

4.4.7. Proposition. Suppose that A and A′ are parental subcomputads of Adj with A ⊆ A′

containing the arrows f , u, and ε. Suppose that T : A→ K is a simplicial functor for whichT (f) = f , T (u) = u, and T (ε) = ε define an adjunction in K2. Then the fibre ET of theisofibration icon(I,K) : icon(A′,K) � icon(A,K) over the vertex T is a contractible Kancomplex.

Proof. The vertices of ET are simply those simplicial functors H : A′ → K which extendthe given simplicial functor T : A → K. It follows, from theorem 4.3.8, that some suchextension does exist and thus that ET is inhabited. Our task is to generalise that argumentand show that if i : X ↪→ Y is any inclusion of simplicial sets then any lifting problem of theform displayed in the displayed left-hand square, or equivalently the composite rectangle

X� _

i

��

// ET //

��

icon(A′,K)

icon(I,K)��

Y!//

??

∆0

T// icon(A,K)


has a solution as illustrated by the dashed map. Transposing, we obtain an equivalentlifting problem

AT //

� _

I��

K K!// KYKi��

A′ // KX

(4.4.8)

in the category of simplicial categories.Now the vertical simplicial functor on the right of this diagram is a local isofibration; its

action on the hom-space from C to D is K(C,D)i : K(C,D)Y � K(C,D)X . Furthermore,as noted in observation 4.3.2, the simplicial functor K! : K → KY preserves the adjunctionsof K2. In particular, the upper horizontal map carries f , u, and ε to the data of anadjunction in KY , and we may apply theorem 4.3.8 to provide us with a solution to thetransformed lifting problem (4.4.8) as required. �

Our first homotopical uniqueness result arises by specialising proposition 4.4.7 to thecase where A = {ε} and A′ = Adj.

4.4.9. Observation. We use the universal property of icon({ε},K) to deduce a new descrip-tion of this space. Observation 4.4.1 tells us that simplicial maps X → icon({ε},K) are inbijective correspondence with simplicial functors T : {ε} → KX . Any such T is completelyand uniquely determined by giving objects A and B of K, 0-arrows T (f) ∈ K(B,A)X

and T (u) ∈ K(A,B)X , and a 1-arrow T (ε) : T (f)T (u) ⇒ idA ∈ K(A,A)X . This amountsto giving a triple of simplicial maps T (f) : X → K(B,A), T (u) : X → K(A,B), andT (ε) : X → K(A,A)2 which make the following square

XT (ε)

//

(!,T (f),T (u))

��

K(A,A)2

(p1,p0)��

∆0 ×K(B,A)×K(A,B)idA×◦B

// K(A,A)×K(A,A)

commute. More concisely, we can express all of this information as a single map from Xinto the pullback of the diagram formed by the right hand vertical and lower horizontalmaps of this square. On consulting I.3.3.15 or Example 5.1.9 below, we recognise that thispullback is precisely the comma quasi-category ◦B ↓ idA displayed in the following diagram:

◦B ↓ idAp1

yyyy ψ⇐∆0

idA %%

K(B,A)×K(A,B)## ##

p0

◦B{{

K(A,A)

In conclusion, the space icon({ε},K) is isomorphic to the (possibly large) coproduct, in-dexed by pairs A,B ∈ objK, of comma quasi-categories ◦B ↓ idA.

48 RIEHL AND VERITY

4.4.10. Definition (the space of counits). An object of

icon({ε},K) ∼=∐

A,B∈objK

◦B ↓ idA

may be written as a triple (f, u, ε) where f ∈ K(B,A) and u ∈ K(A,B) are 0-arrows andε : fu⇒ idA ∈ K(A,A) is a 1-arrow. We are most interested in those objects (f, u, ε) withthe property that ε represents the counit of an adjunction f a u in K2; we write (f a u, ε)to denote an object satisfying that condition.

Now define counit(K) to be the simplicial subset of icon({ε},K) whose 0-simplices arethe objects (f a u, ε), whose 1-simplices are the isomorphisms between them, and whosehigher simplices are precisely those whose vertices and edges are members of these classes.As a quasi-category whose 1-simplices are isomorphisms, counit(K) is a Kan complex,which we call the space of counits in K.

Any given homotopy coherent adjunction H : Adj → K provides us with an adjunctionH(f) a H(u) in K2 with counit represented by the 1-arrow H(ε); it follows that H restrictsalong the inclusion I : {ε} ↪→ Adj to give an object (H(f) a H(u), H(ε)) of counit(K).By lemma 4.4.6, all of the arrows of cohadj(K) are isomorphisms; therefore, as functorspreserve isomorphisms, every arrow in cohadj(K) maps to an arrow of counit(K) underthe isofibration icon(I,K) : icon(Adj,K) � icon({ε},K). Consequently this map factorsthrough the space of counits to give an isofibration pC : cohadj(K) � counit(K) of Kancomplexes.

An object H : Adj → K of cohadj(K) is in the fibre Eε of the isofibration cohadj(K) �counit(K) over some (f a u, ε) precisely if it is a lift of the adjunction f a u to a homotopycoherent adjunction which happens to map ε to the chosen counit representative ε. We callthis fiber the space of homotopy coherent adjunctions extending the counit ε. Proposition4.4.7 specialises to prove our first uniqueness theorem:

4.4.11. Theorem. The space Eε of homotopy coherent adjunctions extending the counit εis a contractible Kan complex.

Theorem 4.4.11 has the following extension.

4.4.12. Proposition. The isofibration pC : cohadj(K)� counit(K) is a trivial fibration ofKan complexes.

Proof. The map pC : cohadj(K) � counit(K) is an isofibration between Kan complexes,and hence a Kan fibration. The conclusion follows immediately from theorem 4.4.11 andthe following standard result from simplicial homotopy theory. �

4.4.13. Lemma. A Kan fibration p : E � B is a trivial fibration if and only if its fibresare contractible.

Proof. This can be proven either by appealing to the long exact sequence of a fibration orby a direct combinatorial argument (see [23, 5.4.16] or [20, 17.6.5]). �


It is also natural to ask what happens if we start only with a single 0-arrow f : B →A, which we know has some right adjoint in K2, and consider all of its extensions toa homotopy coherent adjunction. To answer this question we start by considering theisofibration icon(I,K) : icon({ε},K) � icon({f},K) of quasi-categories induced by theinclusion I : {f} ↪→ {ε} of subcomputads of Adj.

4.4.14. Observation. Observation 4.4.1 tells us that simplicial maps X → icon({f},K) arein bijective correspondence with simplicial functors T : {f} → KX . Any such T is com-pletely and uniquely specified by giving the 0-arrow T (f) of KX . This in turn correspondsto specifying a pair of objects A and B of K and a simplicial map T (f) : X → K(B,A)or, in other words, to giving a simplicial map T (f) : X → ∐

A,B∈objKK(B,A). Thus, thespace icon({f},K) is isomorphic to the (possibly large) coproduct

∐A,B∈objKK(B,A) of

hom-spaces.Combined with observation 4.4.9, we see that the isofibration icon(I,K) : icon({ε},K)�

icon({f},K) is isomorphic to the coproduct of the family of projection isofibrations

◦B ↓ idAp0 // // K(B,A)×K(A,B)

π0 // // K(B,A)

indexed by pairs of objects A,B ∈ K. In particular, the fibre of this isofibration over anobject f ∈ K(B,A) is isomorphic to the comma quasi-category K(A, f) ↓ idA.

4.4.15. Definition (the space of left adjoints). Define leftadj(K) to be the simplicial subsetof icon({f},K) ∼=

∐A,B∈objKK(B,A) whose 0-simplices are those 0-arrows f of K which

possess a right adjoint in K2, whose 1-simplices are the isomorphisms between them, andwhose higher simplices are precisely those whose vertices and edges are members of theseclasses. As a quasi-category whose 1-simplices are isomorphisms, leftadj(K) is a Kancomplex, which we call the space of left adjoints in K.4.4.16. Observation. An object (f a u, ε) of counit(K) maps to the object f of leftadj(K)

under the isofibration icon(I,K) : icon({ε},K) � icon({f},K), which restricts to give anisofibration qL : counit(K)� leftadj(K) of Kan complexes.

4.4.17. Proposition. The isofibration qL : counit(K) � leftadj(K) is a trivial fibration ofKan complexes.

Proof. An isofibration of Kan complexes is a Kan fibration, so this result follows fromlemma 4.4.13 once we show that the fibres of qL are contractible.

If f ∈ K is an object of leftadj(K) then the fibre Ff of qL : counit(K) � leftadj(K)over f is isomorphic to a sub-quasi-category of the fibre K(A, f) ↓ idA of the isofibrationicon(I,K) : icon({ε},K) � icon({f},K). Its objects are pairs (u, ε) which have the prop-erty that u is right adjoint to the fixed 0-arrow f with counit represented by the 1-arrowε : fu⇒ idA.

Given such an object (u, ε) then we may apply the simplicial functor K(A,−) to the ad-junction f a u to obtain an adjunction of quasi-categories K(A, f) a K(A, u) : K(A,A)→

50 RIEHL AND VERITY

K(A,B) whose counit is represented by K(A, ε). By I.4.4.8, the object (u, ε) is a terminalobject in K(A, f) ↓ idA. Therefore, the fibre Ff is simply the full sub-quasi-category ofterminal objects in K(A, f) ↓ idA and, as such, it is contractible as required. �

Composing the trivial fibrations of propositions 4.4.12 and 4.4.17, we obtain a trivialfibration pL : cohadj(K) � leftadj(K) which maps each homotopy coherent adjunctionH : Adj → K to its left adjoint 0-arrow H(f). So if f ∈ K(B,A) is a left adjoint 0-arrow, that is to say an object in leftadj(K), then the fibre Ef of pL over that vertexhas objects which are precisely those homotopy coherent adjunctions H : Adj → K forwhich H(f) = f . Consequently, we call Ef the space of homotopy coherent adjunctionsextending the left adjoint f . Now lemma 4.4.13 applied to the trivial fibration pL tells usthat such extensions exist and are homotopically unique, in the sense that the fibre Ef isa contractible Kan complex, proving our second uniqueness theorem.4.4.18. Theorem. The space Ef of homotopy coherent adjunctions extending the left ad-joint f is a contractible Kan complex. �

5. Weighted limits in qCat∞Of paramount importance to enriched category are the notions of weighted limit and

weighted colimit. Here we consider only three sorts of enrichment — in sets, in categories,or in simplicial sets — so we may as well suppose that the base for enrichment is a completeand cocomplete cartesian closed category V .

Our aim in §5.2 is to show that qCat∞ admits a large class of weighted limits: thosewhose weights are projective cofibrant simplicial functors. These will be used to develop a“formal” theory of monads in the quasi-categorical context by extending a new presentationof the analogous 2-categorical results. For the reader’s convenience, we review the basicsof the theory of weighted limits in §5.1. A more thorough treatment can be found in[14] or [22]. In §5.3, we establish a correspondence between projective cofibrant simplicialfunctors and certain relative simplicial computads that will be exploited in section 6 toidentify projective cofibrant weights.

5.1. Weighted limits and colimits. An ordinary limit is an object representing theSet-valued functor of cones over a fixed diagram. But in the enriched context, this Set-based universal property is insufficiently expressive. The intuition is that in the presenceof extra structure on the hom-sets of a category, cones over a diagram might come in exotic“shapes.”5.1.1. Definition (cotensors). For example, in the case of a diagram of shape 1 in a V-categoryM, the shape of a cone might be an object V ∈ V . Writing D ∈M for the objectin the image of the diagram, the V -weighted limit of D is an object V t D ∈M satisfyingthe universal property

M(M,V t D) ∼= V(V,M(M,D))

where this isomorphism is meant to be interpreted in the category V . For historical reasons,V t D is called the cotensor ofD ∈M by V ∈ V . Assuming the objects with these defininguniversal properties exist, cotensors define a bifunctor − t − : Vop ×M→M.


For example, a closed symmetric monoidal category is always cotensored over itself: thecotensor is simply the internal hom. The cotensor V t D is also denoted by DV when thecontext disambiguates between objects D ∈M and V ∈ V .5.1.2. Definition (weighted limits). Suppose A and M are respectively small and largeV-categories and writeMA for the category of V-functors and V-natural transformations.Suppose further that M is complete and admits cotensors. Then the weighted limit bi-functor { , }A : (VA)op ×MA →M is defined by the formula

{W,D}A :=

∫a∈A

W (a) t D(a)

:= lim

(∏a∈A

W (a) t D(a)⇒∏a,b∈A

(A(a, b)×W (a)) t D(b)

) (5.1.3)

Here, the weightW for the limit of a diagram D of shape A is a covariant V-valued functorof A. We refer to the object {W,D}A as the limit of the diagram D weighted by W . It ischaracterised by the universal property

M(M, {W,D}A) ∼= VA(W,M(M,D)) (5.1.4)where the isomorphism is again interpreted in V .

A map of weights V → W induces a functor between the weighted limits {W,D}A →{V,D}A which we refer to as the functor derived from the map V → W .

5.1.5. Example (representable weights). Let Aa denote the covariant V-enriched repre-sentable of A at an object a. The bifunctor (5.1.3) admits canonical isomorphisms

{Aa, D}A ∼=∫b∈A

A(a, b) t D(b) ∼= D(a) (5.1.6)

which are natural in a ∈ A and D ∈ MA; this result is simply a recasting of the clas-sical Yoneda lemma. Hence, limits weighted by representables are computed simply byevaluating the diagram at the appropriate object.

5.1.7. Observation. The defining universal property of the weighted limit bifunctor, asexpressed in the natural isomorphism of (5.1.4), provides us with an enriched adjunction

(VA)op

{−,D}A

22⊥ MM(−,D)

rr

for each fixed diagramD. Consequently, the right adjoint functor {−, D}A carries (weighted)colimits in VA to (weighted) limits inM; weighted colimits in VA are simply weighted lim-its in (VA)op. In summary, the weighted limit bifunctor is cocontinuous in the weights. Itfollows, in particular, that weights can be “made-to-order” using colimits; that is a weightconstructed as a colimit of representables will stipulate the expected universal property.

52 RIEHL AND VERITY

5.1.8. Example (diagrams). The category VA of weights admits pointwise tensors byobjects of V , satisfying a universal property dual to that of definition 5.1.1. As V iscartesian monoidal, we adopt the notation V ×W for the tensor of V ∈ V by W ∈ VA

defined by (V ×W )(a) := V ×W (a).The cocontinuity of the weighted limit bifunctor in its first variable tells us that if D is a

diagram inMA and V is an object in V then the weighted limit {V ×W,D}A is isomorphicto V t {W,D}A. So, in particular, it follows from example 5.1.5 that {V × Aa, D}A isnaturally isomorphic to V t D(a). Particularly in the quasi-categorical context appearingbelow, the cotensor V t D(a) is often referred to as the object of diagrams of shape V inD(a).

5.1.9. Example (comma quasi-categories). Let A be the category • → • ← •. Let W bethe sSet-valued weight with this shape whose image is

∆0 δ0

−→ ∆1 δ1

←− ∆0

The limit of a diagram D

Bf−→ A

g←− C

in the simplicial category qCat∞ ↪→ sSet weighted by W is the comma quasi-category f ↓gintroduced in I.3.3.15, constructed by the pullback

f ↓ g

��

// A2

��

C ×Bg×f// A× A.

5.1.10. Example (homotopy limits as weighted limits). The homotopy limit of a diagramof shape A taking values in the fibrant objects of a simplicial model category is the limitweighted by A/− : A→ sSet. This is the Bousfield Kan formula [4].

5.1.11. Lemma (weighted limits and Kan extensions). Suppose given a V-functor K : D→C, a diagram D : C →M, and a weight W : D → V. The limit of the restricted diagramDK weighted by W is isomorphic to the limit of D weighted by the left Kan extension ofW along K.

{W,DK}D ∼= {lanKW,D}C

ClanKW

��⇑∼=

DW

//

K>>

V

Proof. The defining universal properties of these weighted limits are easily seen to coincide;see [14, 4.57]. �


5.2. Weighted limits in the quasi-categorical context. The following propositionshows that the limit of any diagram of quasi-categories weighted by a projective cofibrantfunctor is again a quasi-category. The proof is very simple and indeed related conclusionshave been drawn elsewhere; see for instance [8]. For the duration of this section and thenext we shall assume that A is a small simplicial category.

5.2.1. Definition (projective cofibrations). A natural transformation in sSetA is said tobe a projective cofibration if and only if it has the left lifting property with respect to thosenatural transformations which are pointwise trivial fibrations. The maps in the set

{∂∆n ×Aa → ∆n ×Aa | n ≥ 0, a ∈ A}of projective cells are projective cofibrations. A natural transformation in sSetA is a point-wise trivial fibration if and only if it has the right lifting property with respect to allprojective cells. We say that a natural transformation i : V → W is a relative projectivecell complex if it is a countable composite of pushouts of coproducts of projective cells,or equivalently, if it is a transfinite composite of pushouts of projective cells. A simplicialfunctorW in sSetA is a projective cell complex if the unique natural transformation ∅ → Wis a relative projective cell complex.

By the small object argument, we may factor every natural transformation f : V → W insSetA as a composite of a relative projective cell complex i : V → U followed by a pointwisetrivial fibration p : U → W . It follows that a map i : V → W is a projective cofibration ifand only if it is a retract of a relative projective cell complex.

5.2.2. Proposition. Let i : V → W be a projective cofibration of weights in sSetA, and sup-pose that p : D → E is a natural transformation of diagrams in sSetA and a pointwise (triv-ial) fibration in the Joyal model structure. Then the Leibniz limit map {i, p}∧A : {W,D}A →{W,E}A ×{V,E}A {V,D}A is also a (trivial) fibration.

Proof. The natural transformation i : V → W is a projective cofibration if and only if itis a retract of a countable composite of pushouts of coproducts of projective cells. Obser-vation 5.1.7 tells us that the weighted limit bifunctor is cocontinuous in its first variable,so the Leibniz map {i, p}∧A as a retract of a countable tower of pullbacks of products ofLeibniz maps of the form:{in ×Aa, p}∧A : {∆n ×Aa, D}A −→ {∆n ×Aa, E} ×{∂∆n×Aa,E} {∂∆n ×Aa, D} (5.2.3)

Hence, it suffices to show that each of these is a (trivial) fibration. Example 5.1.8 providesa natural isomorphism {X ×Aa, D}A ∼= D(a)X from which we see that the Leibniz map(5.2.3) is isomorphic to the Leibniz hom:

hom(in, pa) : D(a)∆n −→ E(a)∆n ×E(a)∂∆n D(a)∂∆n

Now pa : D(a) → E(a) is a (trivial) fibration, because p is a pointwise (trivial) fibrationby assumption, and so (5.2.3) is also a (trivial) fibration as a consequence of the fact thatthe Joyal model structure is cartesian. �

54 RIEHL AND VERITY

5.2.4. Proposition. The full simplicial subcategory qCat∞ of quasi-categories is closed insSet under limits weighted by projective cofibrant weights in the sense that qCat∞ has andqCat∞ ↪→ sSet preserves such limits.

Proof. Since qCat∞ is a full simplicial subcategory of sSet, all we need do is show that ifW : A→ sSet is a projective cofibrant weight andD : A→ sSet is a diagram whose verticesare all quasi-categories then the weighted limit {W,D}A in sSet is also a quasi-category.This is a special case of proposition 5.2.2. �

5.2.5. Remark. Simplicial limits with projective cofibrant weights should be thought of asanalogous to flexible 2-limits, i.e., 2-limits built out of products, inserters, equifiers, andretracts (splittings of idempotents) [2]. The flexible limits also include iso-inserters, descentobjects, and comma objects. When a 2-category A is regarded as a simplicial category, thechange-of-base functor h∗ : sSetA → CatA carries projective cofibrant weights to flexibleweights. The weights for flexible limits are the cofibrant objects in a model structure onthe diagram 2-category CatA that is enriched over the folk model structure on Cat. Inanalogy with our result, the fibrant objects in a Cat-enriched model structure are closedunder flexible weighted limits [15, theorem 5.4].

The next result shows that limits weighted by projective cofibrant weights are homo-topical, that is, preserve pointwise equivalences between diagrams in qCat∞.

5.2.6. Proposition. Let W : A → sSet be projective cofibrant, and let D,E : A → qCat∞be a pair of diagrams equipped with a natural transformation w : D → E which is a point-wise equivalence. Then the induced map {W,D}A → {W,E}A is an equivalence of quasi-categories.

Proof. Applying the construction of Ken Brown’s lemma, w : D → E may be factored asthe composite of a right inverse to a pointwise trivial fibration followed by a pointwisetrivial fibration. So it suffices to show that if w : D → E is a pointwise trivial fibrationthen {W,D}A → {W,E}A is an equivalence. This is a special case of proposition 5.2.2. �

5.2.7. Remark. The proofs of propositions 5.2.2, 5.2.4, and 5.2.6 apply mutatis mutandis toshow that the fibrant objects in any model category that is enriched over either Quillen’sor Joyal’s model structure on sSet is closed under weighted limits with projective cofibrantweights and that these constructions are homotopical. The essential input in all cases isthe closure property of the (trivial) fibrations in such model categories with respect toLeibniz cotensors by monomorphisms of simplicial sets.

5.2.8. Example (homotopy limits of quasi-categories). For any small category A, theweightA/− : A→ sSet is projective cofibrant [9, 14.8.5]. AnA-diagram of quasi-categoriescan be regarded as a functor D : A → msSet taking values in the fibrant objects ofthe marked model structure of I.2.3.8. The advantage of this interpretation is that themarked model structure is a simplicial model structure [18, 3.1.4.4]. By the last remarkthe weighted limit of a diagram of naturally marked quasi-categories is again a naturallymarked quasi-category. In this way, we see that qCat∞ is closed under the formation ofhomotopy limits. See [22] for more details.


5.2.9. Remark (2-categorical weighted limits and quasi-categorical weighted limits). Recallour convention to regard a 2-functor W : A→ Cat as a simplicial functor W : A→ qCat∞via the embedding 2-Cat ↪→ sSet-Cat. It follows from the defining weighted limit formula(5.1.3) and the fact that the nerve preserves exponentials that the (2-)limit of a 2-functorD : A→ Cat weighted byW , when regarded as a quasi-category, is isomorphic to the limitof the associated simplicial functor D : A → qCat∞ weighted by the simplicial functorW . Many of the weights appearing in sections 6 and 7 are simplicial re-interpretations of2-functors. In this way, the special case of the quasi-categorical monadicity theorem, inwhich the quasi-categories in question are ordinary categories, can be interpreted directlyin the full subcategory Cat of the simplicial category qCat∞.

5.3. The collage construction. To apply proposition 5.2.4, it will be useful to knowthat certain weights constructed from simplicial computads are projective cofibrant. Thisfollows from a recognition principle which relates projective cofibrations to retracts ofrelative simplicial computads between collages.

5.3.1.Definition (the collage construction). LetW : A→ sSet be a simplicial functor. Thecollage of W is a simplicial category collW containing A as a full simplicial subcategoryand precisely one additional object ∗ whose endomorphism space is a point. Declare thehom-spaces from a ∈ A to ∗ to be empty and define

collW (∗, a) := W (a).

The action maps A(a, b) × W (a) → W (b) provide the required compositions betweenthese hom-spaces derived from W and the hom-spaces in the full subcategory A. Thisconstruction is functorial: it carries a natural transformation f : V → W to a simplicialfunctor coll(f) : collV → collW which acts as the identity on the copies of A and ∗ inthose collages and whose actions coll(f) : collV (∗, a) → collW (∗, a) are the componentsfa : V (a)→ W (a).

5.3.2. Observation (a right adjoint to the collage construction). For our purposes the collageconstruction is simply a functor of ordinary, unenriched categories. By definition, thefunctor coll(f) : collV → collW associated with a natural transformation f : V → Wcommutes with the inclusions of the coproduct A+ {∗} into the collages that comprise itsdomain and codomain. So we may write the collage construction as a functor from sSetA

to the slice category (A + {∗})/sSet-Cat which carries a simplicial functor W : A → sSetto the inclusion A + {∗} ↪→ collW . This functor admits a right adjoint

(A + {∗})/sSet-Catwgt

22⊥ sSetAcoll

rr

carrying an object F : A + {∗} → E to the simplicial functor wgt(E, F ) : A→ sSet whoseaction on objects is given by wgt(E, F )(a) := E(F (∗), F (a)) and whose action on hom-spaces is determined by composition in E as follows:

A(a, b)× E(F (∗), F (a))F×id

// E(F (a), F (b))× E(F (∗), F (a))◦ // E(F (∗), F (b)).

56 RIEHL AND VERITY

The unit of this adjunction is an isomorphism, implying that the collage construction is afully faithful functor.

A simplicial category W is of the form coll(W ) if and only if it is comprised of a fullsimplicial subcategory isomorphic to A plus one other object ∗ satisfying the conditionsthat W(∗, ∗) = ∆0 and W(a, ∗) = ∅ for all objects a in A.

The following result characterises relative projective cell complexes.

5.3.3. Proposition. A natural transformation i : V → W in sSetA is a relative projec-tive cell complex if and only if its collage coll(i) : collV → collW is a relative simplicialcomputad.

Proof. Exploiting the adjunction coll a wgt, there is a bijective correspondence betweensimplicial functors F : coll(X×Aa)→ B and natural transformationsX×Aa → wgt(B, F I),where I : A + {∗} ↪→ coll(X ×Aa) denotes the canonical inclusion. Applying the definingproperty of the cotensor X×Aa and Yoneda’s lemma, we see that maps of this latter kindcorrespond to simplicial maps X → wgt(B, F I)(a) = B(F (∗), F (a)). Consequently we ob-tain a natural bijective correspondence between simplicial functors F : coll(X ×Aa)→ B

and pairs of simplicial functors F : A → B and F : 2[X] → B with the property thatF (1) = F (a). That latter pair is obtained from F : coll(X × Aa) → B by restrict-ing it to the subcategory A and by composing it with a canonical comparison functorKX : 2[X] → coll(X × Aa) respectively. Using this characterisation, it is easy to checkthat if f : X → Y is any simplicial map then the square

2[X]KX //

2[f ]��

coll(X ×Aa)

coll(f×Aa)��

2[Y ]KY

// coll(Y ×Aa)

(5.3.4)

is a pushout.Transfinite composites and pushouts are colimits of connected diagrams, so they are

both preserved and reflected by the forgetful functor (A+{∗})/sSet-Cat→ sSet-Cat. Fur-thermore, the collage construction is a fully faithful left adjoint functor so it too preservesand reflects all small colimits. For the “only if” direction, we suppose that i : V ↪→ W isa relative projective cell complex, i.e., that it can be expressed as a transfinite compositeof pushouts of natural transformations of the form ∂∆n × Aa ↪→ ∆n × Aa. On apply-ing the collage construction and projecting into sSet-Cat, we obtain a decomposition ofcoll(i) : collV ↪→ collW as a transfinite composite of pushouts of simplicial functors ofthe form coll(∂∆n ×Aa) ↪→ coll(∆n ×Aa). Composing each of those pushouts with thepushout square (5.3.4), we can also express coll(i) as transfinite composite of pushoutsof simplicial functors of the form 2[∂∆n] ↪→ 2[∆n]. This proves that coll(i) is a relativesimplicial computad.

Conversely for the “if” direction, if coll(i) : coll(V ) ↪→ coll(W ) is a relative simplicialcomputad, then it can be expressed as a transfinite composite of functors Iβ : Wβ ↪→Wβ+1

each of which is a pushout of a functor 2[∂∆n] ↪→ 2[∆n] for some n ≥ 0; because coll(V )


and coll(W ) share the same sets of objects, we will not require the functor ∅ ↪→ 1 ofdefinition 2.1.4.

If β ≤ β′ then we may regard Wβ as being a simplicial subcategory of Wβ′ . In particularsince each Wβ sits as a simplicial subcategory between coll(V ) and coll(W ) it follows thatit too must have A as a full simplicial subcategory plus one extra object ∗ for whichWβ(∗, ∗) = ∆0 and Wβ(a, ∗) = ∅. Applying the characterisation of observation 5.3.2,there exists an essentially unique simplicial functorW β in sSetA such that coll(W β) ∼= Wβ.As the collage construction is fully faithful, we also obtain induced maps iβ,β′ : W β → W β′

whose images under the collage construction are isomorphic to the connecting functorsIβ,β

′: Wβ ↪→Wβ′ under the chosen isomorphisms coll(W β) ∼= Wβ. Finally, the transfinite

sequence iβ,β′ : W β → W β′ in sSetA is actually a transfinite composite because it mapsunder the collage construction to a sequence which is isomorphic to our chosen transfinitecomposite Iβ,β′ : Wβ ↪→Wβ′ in sSet-Cat and these colimits are reflected.

All that remains is to show that each iβ : W β → W β+1 is a pushout of a projectivecell. We know that coll(iβ) : collW β ↪→ collW β+1 is isomorphic to Iβ : Wβ ↪→Wβ+1, byconstruction, and that latter functor is a pushout of some inclusion 2[∂∆n] ↪→ 2[∆n], soit follows that there is a pushout:

2[∂∆n]Fβ //

� _

��

collW β� _

coll(iβ)��

2[∆n] // collW β+1

However, collW β and collW β+1 can only differ in hom-spaces whose domains are ∗ andwhose codomains are objects of A so the attaching simplicial functor F β must map 0 tothe object ∗ and 1 to some object a of A. It follows we may apply the observations ofthe first paragraph of this proof to factor F β as a composite of the canonical comparisonK∂∆n : 2[∂∆n]→ coll(∂∆n×Aa) and a simplicial functor F β : coll(∂∆n×Aa)→ collW β.As the collage construction is fully faithful, there must exist a unique natural transforma-tion fβ : ∂∆n×Aa → W β with the property that coll(fβ) = F β. This defines a factorisationof the pushout above through the pushout (5.3.4) to give the following diagram

2[∂∆n]K∂∆n //

� _

��

coll(∂∆n ×Aa)� _

coll(i×Aa)

��

coll(fβ)// collW β

� _

coll(iβ)��

2[∆n]K∆n

// coll(∆n ×Aa)coll(gβ)

// collW β+1

in which the right hand square is a pushout by the usual cancellation argument. As thecollage construction reflects pushouts, we conclude that iβ : W β ↪→ W β+1 is a pushout ofthe projective cell i×Aa : ∂∆n ×Aa ↪→ ∆n ×Aa as required. �

Sections 6 and 7 make substantial use of the following special case of proposition 5.3.3.

58 RIEHL AND VERITY

5.3.5. Proposition. A simplicial functor W : A→ sSet is a projective cell complex if andonly if the canonical inclusion A ↪→ collW is a relative simplicial computad.

Proof. Proposition 5.3.3 tells us that ∅ → W is a projective cofibration if and only ifcoll ∅ → collW is a relative simplicial computad. This latter functor is simply the canonicalinclusion A + {∗} ↪→ collW , which is a relative simplicial computad if and only if A ↪→collW is such. �

6. The formal theory of homotopy coherent monads

Let Mnd denote the full sub-2-category of Adj on the object +. We know from corol-lary 3.3.5 and remark 3.3.8 that the hom-category of Mnd is the category �+ and that itshorizontal composition is given by the join operation. So Lawvere’s characterisation [17]of �+ as the free monoidal category containing a monoid tells us that Mnd is the free2-category containing a monad.

We have seen that any adjunction in the 2-category K2 of a quasi-categorically enrichedcategory K extends to a homotopy coherent adjunction, a simplicial functor Adj→ K. Thecomposite simplicial functor Mnd→ Adj→ K is the homotopy coherent monad generatedby that adjunction. More generally, we regard any simplicial functor Mnd → K as ahomotopy coherent monad in K. In this section and the next we justify this definitionby developing the theory of homotopy coherent monads in the quasi-categorical context,including Beck’s monadicity theorem.

The “formal theory of monads” plays homage to Ross Street’s paper [29], which developsa formal 2-categorical theory of monads and their associated Eilenberg-Moore and Kleisliconstructions. However, our method here is not a direct generalisation of his. For example,he defines the Eilenberg-Moore object associated with a monad in a 2-category using auniversal property which is expressed in terms of the associated 2-category of monads andmonad morphisms. In the quasi-categorical context, we will describe the Eilenberg-Mooreobject, which we refer to as the quasi-category of algebras , as a limit of the homotopycoherent monad Mnd → qCat∞ weighted by a projective cofibrant weight extracted fromthe simplicial computad Adj.

This weight for Eilenberg-Moore objects was first observed in the 2-categorical contextby Lawvere [17], but he does not appear to have recognised the connection with the freeadjunction. Our description and analysis of Adj will allow us to take his insights much fur-ther, applying them directly to understanding monadicity in the quasi-categorical context.One novelty in our approach is that we describe almost all constructions and computationsinvolved our proof of Beck’s theorem in terms of the properties of weights of various kindsand of natural transformations between them. This is the topic of section 7.

A key technical point is that the weights we derive from the simplicial category Adj areall shown to be projective cofibrant, as an immediate consequence of the fact that Adj isa simplicial computad. It follows then, by proposition 5.2.4, that limits of diagrams ofquasi-categories weighted by such weights are again quasi-categories. In particular, ourweighted limits approach produces explicit quasi-categorical models of all key structuresinvolved in the theory of homotopy coherent monads.


In §6.1, we introduce the weights for the quasi-category of algebras for a homotopycoherent monad. We then build the associated (monadic) adjunction by exploiting a corre-sponding adjunction of weights. In §6.2, we show that the monadic forgetful functor reflectsisomorphisms. In §6.3, we describe how the objects in the quasi-category of algebras fora homotopy coherent monad are themselves colimits of canonically constructed simplicialobjects. A full proof of this result is deferred to [26], where we prove a substantial gen-eralization that applies to other quasi-categories defined via projective cofibrant weightedlimits, but we outline that argument here.

The proofs in this section are all just formal arguments involving the weights — thequasi-category on which the homotopy coherent monad is defined need not be referenced.In analogy with the classical case, the proof of the quasi-categorical analog of Beck’smonadicity theorem in the next section will require certain hypotheses on the underlyingquasi-categories.

6.1. Weighted limits for the formal theory of monads.

6.1.1. Definition (homotopy coherent monads). A homotopy coherent monad in a quasi-categorically enriched category K is a simplicial functor T : Mnd→ K. The action of thisfunctor on the unique object + in Mnd picks out an object B of K. Its action on the solehom-space �+ of Mnd is given by a functor t : �+ → K(B,B) of quasi-categories which isa monoid map relative to the join operation on �+ and the composition operation on theendo-hom-space K(B,B).

6.1.2. Remark. To fix ideas, from here to the end of the paper we shall work in qCat∞with respect to a fixed homotopy coherent monad T : Mnd → qCat∞ acting on a quasi-category B via a functor t : �+ → BB. However, our arguments can be interpreted equallyin the context of an arbitrary quasi-categorically enriched category K that admits all limitsweighted by projective cofibrant weights.

6.1.3. Observation (weights on Mnd). The constructions that we will apply to homotopycoherent monads will be expressed as limits weighted by projective cofibrant weights insSetMnd. Any simplicial functor W : Mnd → sSet is describable as a simplicial set W =W (+) equipped with a left action · : �+×W → W of the simplicial monoid (�+,⊕, [−1]).Furthermore, a map f : V → W in sSetMnd is a simplicial map which is equivariant withrespect to the actions of �+ on V and W , in the sense that the square

�+ × V�+×f

//

·��

�+ ×W·��

Vf

// W

commutes. In particular, a cone c : W → T (−)A weighted by W over the diagram T withsummit A is specified by giving a simplicial map c : W → BA which makes the following

60 RIEHL AND VERITY

square

�+ ×W·��

t×c// BB ×BA

◦��

W c// BA

commute.

Of course, since homotopy coherent monads are also simplicial functors Mnd→ qCat∞,they too may be expressed as simplicial sets with left �+-actions.

6.1.4. Definition (monad resolutions). Write W+ : Mnd → sSet for the (unique) repre-sented simplicial functor on Mnd. When described as in observation 6.1.3, W+ is thesimplicial set �+ acting on itself on the left by the join operation.

By the Yoneda lemma (5.1.6), the limit of any diagram weighted by a representablesimplicial functor always exists and is isomorphic to the value of the diagram at the rep-resenting object. Hence, the weighted limit {W+, T}Mnd is isomorphic to B, the object onwhich the monad operates, and the data of the limit cone is given by the action t : �+ → BB

of the simplicial functor T on the hom-space of Mnd. We refer to this diagram as themonadresolution and use the following notation for the evident 1-skeletal subset of its image

idB η // tηt //

tη //t2µoo tηt //

ηtt //

ttη //

t3 · · · · · ·tµoo

µtoo

∈ BB.

Evaluating this at any object b we obtain an augmented cosimplicial object in B, whichwe draw as:

b ηb // tbηtb //

tηb //t2bµboo tηtb //

ηttb //

ttηb //

t3b · · · · · ·tµboo

µtboo

∈ B (6.1.5)

When B is an ordinary category, regarded as a quasi-category, remark 5.2.9 applies. Ahomotopy coherent monad is just an ordinary monad and the cone t : �+ → BB is theusual monad resolution.

Another way to describe the weight W+ is to view it as the restriction of the covariantrepresentable Adj+ : Adj→ sSet to the full simplicial subcategory Mnd. This suggests thefollowing definition:

6.1.6.Definition. WriteW− for the restriction of the covariant representable Adj− : Adj→sSet to the full simplicial subcategory Mnd. When described as a left �+-simplicial set, as inobservation 6.1.3, W− has underlying simplicial set �∞ and left action ⊕ : �+×�∞ → �∞.

6.1.7. Definition (quasi-category of algebras). The quasi-category of (homotopy coherent)algebras for a homotopy coherent monad T on an object B in qCat∞ is the weighted limit

B[t] := {W−, T}Mnd.


Once we show that the weight W− is projective cofibrant, proposition 5.2.4 will implythat every homotopy coherent monad possesses an associated quasi-category of algebras.The proof of this fact is entirely straightforward and follows a pattern we shall see repeatedfor other weights below. We identify the collage collW− as a simplicial subcategory ofAdj, use that description to show that it is a simplicial computad, and then appeal toproposition 5.3.5.

6.1.8. Lemma. The simplicial functor W− : Mnd→ sSet is projective cofibrant.

Proof. The collage ofW− can be identified with the (non-full) simplicial subcategory of Adjcontaining the hom-spaces Adj(+,+) and Adj(−,+) but with the hom-spaces from − and+ to − respectively trivial and empty. This is a simplicial computad whose atomic arrowsare precisely those squiggles whose codomain is + and which do not contain any instancesof + in their interiors. Note that the atomic arrows in collW− are not necessarily atomicin Adj, as they may contain any number of occurrences of −, so collW− is not a simplicialsubcomputad of Adj. However, the atomic arrows of Mnd are also those squiggles whichdo not contain any instances of + in their interiors, so Mnd is a simplicial subcomputadof collW−, and it follows that Mnd ↪→ collW− is a relative simplicial computad. Theconclusion now follows from proposition 5.3.5. �

6.1.9. Corollary. Every homotopy coherent monad in qCat∞ admits a quasi-category ofalgebras.

Proof. Immediate from lemma 6.1.8 and proposition 5.2.4. �

6.1.10. Remark. We unpack definition 6.1.7 to view an object of B[t] as a homotopy co-herent algebra. The definition of B[t] as a limit weighted by W− tells us that the objectb : ∆0 → B[t] corresponds to a W−-weighted cone over T with summit ∆0. As discussedin observation 6.1.3, such a cone is simply a simplicial map b : �∞ → B satisfying theequivariance condition that

�+ × �∞⊕��

t×b// BB ×B

ev

��

�∞b

// B

(6.1.11)

commutes. Evaluating b at [0] ∈ �∞ we obtain an object in B, which we shall also denoteby b. Evaluating at σ0 : [1] → [0], we obtain an arrow in B, which we shall denote byβ : tb→ b.

The condition (6.1.11) implies, in particular, that the composite of b : �∞ → B with thefunctor − ⊕ [0] : �+ → �∞ is equal to the resolution displayed in (6.1.5). Drawing thisalgebra in the way that we drew our monad resolutions (6.1.5), we obtain the followingpicture:

b ηb //tbβoo

ηtb //

tηb // t2bµboo

tβoo

ηttb //

tηtb //

ttηb //t3b · · · · · ·

µtboo

tµboo

ttβoo

∈ B (6.1.12)

62 RIEHL AND VERITY

The higher dimensional data of (6.1.12) implies, in particular, that (b, β) defines a h(t)-algebra, in the usual sense, in the homotopy category hB. However, it is not generally thecase that all h(t)-algebras in hB can be lifted to homotopy coherent t-algebras in B.

6.1.13. Example (free monoid monad). Let Kan∞ denote the simplicial category of Kancomplexes. We may construct the free strictly associative monoid on a Kan complex K inthe usual way: take the coproduct

tK :=∐n≥0

Kn

which is again a Kan complex and equip it with the obvious concatenation operation asits product tK × tK → tK. This provides us with a simplicially enriched monad onKan∞ whose monad resolution �+ → KanKan∞

∞ may be transposed to give a left action�+ × Kan∞ → Kan∞ of the strict monoidal category (�+,⊕, [−1]). Here we regard thecategory �+ as being a simplicial category with discrete hom-spaces.

Applying the homotopy coherent nerve construction N : sSet-Cat → sSet, which coin-cides with the usual nerve construction on discrete simplicial categories, we obtain a leftaction �+ ×NKan∞ → NKan∞, transposing to define a monoid map �+ → NKanNKan∞

∞ .This defines a homotopy coherent monad on the (large) quasi-category NKan∞.

Consulting remark 6.1.10, we see that a vertex in the associated quasi-category of coher-ent algebras corresponds to a functor �∞ → NKan∞ satisfying the naturality conditionwith respect to the left actions of �+ on its domain and codomain. We can take the trans-pose of that map under the adjunction C a N to give a simplicial functor C�∞ → Kan∞;hence, a homotopy coherent algebra is a homotopy coherent diagram of shape �∞ inKan∞. The data in the image of this functor picks out a Kan complex K, an action mapβ : tK → K, various composites of these as displayed in (6.1.12), and higher dimensionalhomotopy coherence data that relates those to the structure of the monad resolution atK. In particular, this data ensures that the action map β : tK → K supplies K withthe structure of a strictly associative monoid in the classical homotopy category of Kancomplexes.

Importantly, weighted limits can be used not just to define the quasi-category of algebrasfor a homotopy coherent monad but also the full (monadic) homotopy coherent adjunction.

6.1.14. Definition (monadic adjunction). Composing the Yoneda embedding and the re-striction along Mnd ↪→ Adj, one obtains a simplicial functor

Adjop // sSetAdj // sSetMnd

+ � // W+

− � // W−

(6.1.15)

We know that the weighted limit construction {−, T}Mnd is simplicially contravariantlyfunctorial on the full subcategory of projective cofibrant weights in sSetMnd. In particular,the representable W+ and the weight for quasi-categories of algebras W− are both pro-jective cofibrant, so it follows that we may compose the simplicial functor (6.1.15) with


the weighted limit construction {−, T}Mnd : (sSetMndcof )op → qCat∞ to obtain a homotopy

coherent adjunction Adj→ qCat∞.We denote the primary data involved in this adjunction as follows

{W−, T}Mnd = B[t]ut

22⊥ B ∼= {W+, T}Mnd

f t

rrηt : idB ⇒ utf t

εt : f tut ⇒ idB[t]

and call this the homotopy coherent monadic adjunction associated with the homotopycoherent monad T .

6.2. Conservativity of the monadic forgetful functor.

6.2.1. Definition (conservative functors). We say a functor f : A → B between quasi-categories is conservative if it reflects isomorphisms; that is to say, if it has the propertythat a 1-simplex in A is an isomorphism if and only if its image in B under f is anisomorphism. It is clear that f is conservative if and only if the corresponding functorh(f) : hA→ hB of homotopy categories is conservative.

As in the categorical context, the monadic forgetful functor ut : B[t] → B is alwaysconservative. This will follow from the following general result.

6.2.2. Proposition. Suppose A is a small simplicial category and that i : V ↪→ W in sSetA

is a projective cofibration between projective cofibrant weights with the property that for allobjects a ∈ A the simplicial map ia : V (a) ↪→ W (a) is surjective on vertices. Then for anydiagram D in qCatA∞, the functor {i,D}A : {W,D}A → {V,D}A is conservative.

Proof. Applying the small object argument for the restricted set of projective cells ∂∆n ×Aa ↪→ ∆n × Aa with n > 0, we factor i : V ↪→ W as a composite of a natural trans-formation i′ : V ↪→ U which is a transfinite composite of pushouts of those cells and anatural transformation p : U → W which has the right lifting property with respect tothose cells. This second condition means that for all objects a of A the simplicial mappa : U(a)→ W (a) has the right lifting property with respect to each ∂∆n ↪→ ∆n for n > 0.Now ia = pai

′a is surjective on vertices, by assumption, so it follows that pa is also surjec-

tive on vertices. This means that p has the right lifting property with respect to each cell∅ ∼= ∂∆0 ×Aa ↪→ ∆0 ×Aa

∼= Aa, so pa is actually a trivial fibration. As i is a projectivecofibration by assumption, it has the right lifting property with respect to the pointwisetrivial fibration p. Solving the obvious lifting problem between i and p, we conclude that iis a retract of i′. This demonstrates that i is a retract of a transfinite composite of pushoutsof the restricted set of projective cells {∂∆n ×Aa ↪→ ∆n ×Aa | a ∈ A, n > 0}.

Arguing as in the proof of proposition 5.2.2, we may now express {i,D}A as a retract ofa transfinite co-composite (limit of a tower) of pullbacks of functors of the form

D(a)i : D(a)∆n −→ D(a)∂∆n

with n > 0. The class of conservative functors is closed under transfinite co-composites,pullbacks, and splitting of idempotents: working in the marked model structure, a functoris conservative if and only if it has the right lifting property with respect to 2 ↪→ 2]. So

64 RIEHL AND VERITY

it is enough to show that each of those functors is conservative. This is an easy corollaryof lemma I.2.3.10: pointwise equivalences in D(a)∆n are detected in D(a)∂∆n , providedn > 0, because ∂∆n → ∆n is surjective on 0-simplices. �

6.2.3. Corollary. The monadic forgetful 0-arrow ut : B[t]→ B is conservative.

Proof. The forgetful functor ut is constructed by applying the contravariant weighted limitfunctor {−, T}A to the natural transformation which arises by applying the simplicial func-tor displayed in (6.1.15) to the 0-arrow u in Adj. In other words, this is the natural transfor-mation W+ ↪→ W− which acts by pre-whiskering the elements of W+(+) = Adj(+,+) withthe 0-arrow u to give an element of W−(+) = Adj(−,+). Our graphical calculus makesclear that this pre-whiskering operation is injective, so it follows that we may use it toidentify collW+ with a simplicial subcategory of collW−, which we have already identifiedwith a simplicial subcategory of Adj in the proof of lemma 6.1.8.

Under this identification collW+ becomes a simplicial subcategory of collW− whichdiffers from it solely to the extent that its hom-space collW+(−,+) contains only thosesquiggles of collW+(−,+) = Adj(−,+) that decompose as au for some unique squigglea in Adj(+,+). In particular, every atomic arrow of collW− is also atomic in collW+.Thus, collW+ ↪→ collW− is a relative simplicial computad to which we may apply proposi-tion 5.3.3 to show that −◦u : W+ ↪→ W− is a relative projective cell complex. Furthermore,every 0-arrow in Adj is an alternating composite of the atomic 0-arrows u and f , so it isclear that every 0-arrow in W−(+) = Adj(−,+) does decompose as au and is thus in theimage of − ◦ u : W+ ↪→ W−. Applying proposition 6.2.2, we conclude that ut : B[t]→ B isconservative. �

6.2.4. Observation. In the proof of corollary 6.2.3, we observed thatW+ ↪→ W− is a relativeprojective cell complex. By proposition 5.3.3, we can extract an explicit presentationfrom the corresponding relative simplicial computad collW+ ↪→ collW− via our graphicalcalculus. Applying the weighted limit {−, T} to the map W+ ↪→ W−, this translates toa presentation of the monadic forgetful functor as a limit of a tower of isofibrations, eachlayer of which is defined as the pullback of a map B∆n

� B∂∆n corresponding to anatomic n-arrow of collW− not in the image of collW+. In particular, ut : B[t] � B is anisofibration.

6.2.5. Remark. We may express the projective cofibration − ◦ u : W+ ↪→ W− in the formof observation 6.1.3 using the representation of Adj given in remark 3.3.8. Under thatinterpretation it is the simplicial map − ⊕ [0] : �+ ↪→ �∞, which satisfies the requiredequivariance condition

�+ × �+

�+×(−⊕[0])//

⊕��

�+ × �∞⊕��

�+ −⊕[0]// �∞

as an immediate consequence of the associativity of the join.


6.3. Colimit representation of algebras. Perhaps the key technical insight enablingBeck’s proof of the monadicity theorem is the observation that any algebra is canonicallya colimit of a particular diagram of free algebras. More precisely any algebra (b, β) for amonad t on a category B is a ut-split coequaliser

t2bµb //

tβ //tbtηboo β //

ηtboo

bηboo

(6.3.1)

Here the solid arrows are maps which respect t-algebra structures on these objects, whereasthe dotted splittings are not. Split coequalisers are examples of absolute colimits, thatis to say colimits which are preserved by any functor. In particular they are preservedby t : B → B itself, a fact we may exploit in order to show that the forgetful functorut : B[t]→ B creates creates the canonical colimits of the form (6.3.1).

On our way to a monadicity theorem that can be applied to homotopy coherent ad-junctions of quasi-categories, we demonstrate that any vertex in the quasi-category B[t] ofalgebras for a homotopy coherent monad T : Mnd→ qCat∞ has an analogous colimit pre-sentation. In this context, the ut-split coequaliser (6.3.1) is replaced by a canonical ut-splitaugmented simplicial object. In this section, we give a precise statement of this result anda sketch of its proof. The full details are deferred to [26] because the argument, relying onour description of the quasi-category of algebras as a projective cofibrant weighted limit,applies to general quasi-categories defined as limits of this form.

As explained in section I.5, colimits in a quasi-category are encoded by absolute leftlifting diagrams in qCat2 of a particular form. For the reader’s convenience, we brieflyrecall definition I.5.2.9.I.5.2.9. Definition (colimits in a quasi-category). We say a quasi-category A admits col-imits of a family of diagrams k : K → AX of shape X if there is an absolute left liftingdiagram in qCat2

⇑λ

A

c��

Kk//

colim

==

AX

in which c, the “constant map”, is the adjoint transpose of the projection πA : A×X → A.For example, the split augmented simplicial objects in a quasi-category B provide us

with a family of colimit diagrams:I.5.3.1. Theorem. For any quasi-category B, the canonical diagram

⇑

B

c��

B�∞

ev0

;;

res// B�op

(6.3.2)

is an absolute left lifting diagram. Hence, given any simplicial object admitting an aug-mentation and a splitting, the augmented simplicial object defines a colimit cone over theoriginal simplicial object. Furthermore, such colimits are preserved by any functor.

66 RIEHL AND VERITY

6.3.3. Recall (constructing the triangle in theorem I.5.3.1). The object [−1] is terminal inthe category �op

+ , so there exists a necessarily unique natural transformation:

�op � � //

!

⇓

�op+

1[−1]

>>(6.3.4)

Furthermore, post-composition by the arrow u : − → + of Adj provides us with an em-bedding Adj(−,−) ↪→ Adj(−,+), which we denote by u ◦ − : �op

+ ↪→ �∞. On applying thecontravariant 2-functor B(−) to this data we obtain the following diagram

⇑

B

c��

B�∞Bu◦−

// B�op+

ev−1

99

res// B�op

=:⇑

B

c��

B�∞

ev0

99

res// B�op

whose composite is the triangle displayed in the statement of theorem I.5.3.1.

The importance of this particular family of colimits is that their presence in B allows usto infer the existence of a more general family of colimits in the quasi-category B[t]. Thefollowing definition specifies the class of diagrams in that family:

6.3.5. Definition (u-split augmented simplicial objects). By remark 3.3.8, the image ofthe embedding u ◦ − : �op

+ ↪→ �∞ of recollection 6.3.3 is the subcategory of �∞ generatedby all of its elementary operators except for the face operators δ0 : [n − 1] → [n] for eachn ≥ 1. We call these extra face maps δ0 splitting operators.

Given any functor u : C → B of quasi-categories, a u-split augmented simplicial object isan augmented simplicial object �op

+ → C that, when mapped to B by u, comes equippedwith an extension �∞ → B, providing actions of the splitting operators and associatedhigher coherence data. In other words, such structures comprise pairs of horizontal functorsin the following diagram

�op+� _

u◦−��

// C

u

��

�∞ // B

which make that square commute.We may define a quasi-category S(u) of u-split augmented simplicial objects by forming

the following pullback:S(u)

��

// B�∞

Bu◦−��

C�op+

u�op+

// B�op+

(6.3.6)


This is a pullback of exponentiated quasi-categories whose right hand vertical is an isofi-bration, as u ◦ − : �op

+ ↪→ �∞ is injective. It follows that S(u) is indeed a quasi-category.This construction may also be described as the limit of the diagram 2 → qCat∞ whoseimage is the functor u : C → B weighted by the projective cofibrant weight 2 → sSetwhose image is u ◦ − : �op

+ ↪→ �∞.

The following proposition motivates our consideration of this particular class of diagrams:

6.3.7. Proposition. The monadic forgetful functor ut : B[t]→ B creates colimits of ut-splitsimplicial objects. It follows immediately that ut both preserves and reflects such colimits.

Our proof of this result relies on the following theorem which we prove as corollary 5.5of [26], where it appears as a special case of a much more general theorem proven there:

6.3.8. Theorem. The monadic forgetful functor ut : B[t] → B of a homotopy coherentmonad creates any colimits that t : B → B preserves.

6.3.9. Observation. Before sketching a proof of theorem 6.3.8, let us expand upon its state-ment. It asks us to show that if k : K → B[t]X is a family of diagrams in B[t] whose un-derlying diagrams (ut)Xk : K → BX admit colimits in B that are preserved by t : B → B,then we may lift them to give colimits of the diagrams we started with in B[t].

In other words, consider a family of diagrams k : K → B[t]X and an absolute left liftingdiagram

⇑λ

B

c��

K(ut)Xk

//

colim

::

BX

(6.3.10)

so that the composite diagram

⇑λ

B

c��

t // B

c��

K(ut)Xk

//

colim

::

BX

tX// BX

is again an absolute left lifting diagram. Theorem 6.3.8 asserts that under these assump-tions there is a diagram

⇑λ

B[t]

c��

Kk//

colim==

B[t]X

68 RIEHL AND VERITY

which lies over (6.3.10), in the sense that it satisfies the equality

⇑λ

B[t]

c��

ut // B

c��

Kk//

colim==

B[t]X(ut)X

// BX

= ⇑λ

B

c��

K(ut)Xk

//

colim

::

BX

and that any such diagram λ lying over (6.3.10) in this way is itself an absolute left liftingdiagram.

Sketch proof of theorem 6.3.8. The map ut : B[t]→ B is induced by the inclusion of weightsW+ ↪→ W− applied to the homotopy coherent monad T . As noted in observation 6.2.4,we may extract an explicit presentation of W+ ↪→ W− as a relative projective cell complexfrom our graphical calculus. Passing to weighted limits of the homotopy coherent monadT , ut : B[t]� B is then the limit of a tower of isofibrations defined as pullbacks of maps ofthe form B∆n

� B∂∆n ; these isofibrations arise as the limit of T weighted by a projectivecell W+ × ∂∆n ↪→ W+ ×∆n indexed by an atomic n-arrow of collW− not in the image ofcollW+.

By proposition I.5.2.18, the cotensors of B admit geometric realisations of split aug-mented simplicial objects, defined pointwise in B. Because t preserves these colimits, themaps in the pullback diagrams defining the layers in the tower for ut : B[t] � B preservethese colimits. We conclude by arguing inductively that each pullback admits and the legsof the pullback cone preserve such colimits. The limit stage of this induction creates thedesired colimits in B[t]. �

Proof of proposition 6.3.7. Start by applying the 2-functor B[t](−) to the 2-cell in (6.3.4)to obtain a triangle which we may combine with the canonical projection S(ut)� B[t]�

op+

of (6.3.6) to give a composite triangle:

⇑

B[t]

c��

S(ut) // B[t]�op+

ev−1

88

res// B[t]�

op

(6.3.11)

Now, the 2-functoriality properties of exponentiation provide us with the following pastingequation

⇑

B[t]

c��

ut // B

c��

B[t]�op+

res//

ev−1

::

B[t]�op

(ut)�op// B�op

=⇑

B

c��

B[t]�op+

(ut)�op+// B�op

+

ev−1

99

res// B�op


which we may combine with the defining pullback square (6.3.6) to show that the compositeof the triangle in (6.3.11) with the functor ut reduces to:

⇑

B

c��

S(ut) // B�∞

ev0

;;

res// B�op

(6.3.12)

Here the triangle on the right is simply that given in the statement of theorem I.5.3.1, whoseconstruction is described in recollection 6.3.3. In other words, we have shown that thetriangle in (6.3.11) lies over the one in (6.3.12), in the sense discussed in observation 6.3.9.

The triangle in (6.3.12) is an absolute left lifting diagram, simply because it is obtained bypre-composing the absolute left lifting diagram of theorem I.5.3.1 by the functor S(ut)→B�∞ of (6.3.6). Theorem I.5.3.1 also tells us that these colimits are preserved by allfunctors and so, in particular, they are preserved by t : B → B. Consequently, we mayapply theorem 6.3.8 to show that ut : B[t] → B creates these colimits as postulated. Onconsulting observation 6.3.9, we see that we have succeeded in showing that the particulartriangle given in (6.3.11) is an absolute left lifting diagram which displays those colimits.

�

Proposition 6.3.7 can be specialised to provide the promised representation of homotopycoherent algebras as colimits of diagrams of free such algebras. All that remains to do sois to define this particular family of diagrams of ut-split augmented simplicial objects inB[t].

6.3.13. Observation (a direct description of S(ut)). We can express the quasi-category S(ut)of ut-split simplicial objects in B[t] directly as a limit of the homotopy coherent monadT weighted by a projectively cofibrant weight. To see how this may be achieved, start byrecalling that the forgetful functor ut is constructed by applying the covariant weightedlimit functor {−, T}Mnd to the natural transformation − ◦ u : W+ ↪→ W− described inremark 6.2.5. This features in the following pushout in sSetMnd of weights

W+ × �op+

� _

W+×(u◦−)

��

� �(−◦u)×�op

+// W− × �op

+� _

��

W+ × �∞ ��

// Ws

(6.3.14)

in which the products are tensors of weights by simplicial sets. Now the contravariantsimplicial functor {−, T} is cocontinuous, so it carries this pushout to a pullback which iseasily seen to be the ut instance of the pullback (6.3.6). Consequently, S(ut) is canonicallyisomorphic to the limit {Ws, T}Mnd.

6.3.15. Observation. Every homotopy coherent algebra for T gives rise to a ut-split simpli-cial object in B[t], and this construction may be encapsulated in a functor B[t] → S(ut)which we now describe. Using observation 6.1.3, we re-express the objects and maps thatoccur in the diagram whose pushout we formed in (6.3.14) in terms of the structure of the

70 RIEHL AND VERITY

simplicial category Adj. Doing so we get the upper horizontal and left-hand vertical mapsin the following square

Adj(+,+)× Adj(−,−)(−◦u)×Adj(−,−)

//

Adj(+,+)×(u◦−)��

Adj(−,+)× Adj(−,−)

◦��

Adj(+,+)× Adj(−,+) ◦// Adj(−,+)

whose lower right-hand vertex is the simplicial set underlyingW−. This diagram commutesby associativity of the composition in Adj, and each of the maps respects the manifest left(post-composition) actions of Adj(+,+) = �+ on each of its nodes. Hence, this definesa cone under (6.3.14) which induces an action preserving map Ws → W−, and applying{−, T} to all of this data we obtain a commutative diagram:

B[t]

""

t•

''

s•

��

S(ut)

��

// B[t]�op+

ut

��

B�∞res// B�op

+

(6.3.16)

Finally, we wish to describe the diagrams in the image of the functor t• : B[t]→ B[t]�op+ .

Recall from remark 3.3.8 that the map of weights defining t• is given by the join operation⊕ : �∞ × �op

+ → �∞. The functor t• carries each homotopy coherent algebra b : �∞ → B,presented by the diagram (6.1.12), to a functor b : �op

+ → B[t], which we now describe. Asin remark 6.1.10, the equaliser formula (5.1.3) for weighted limits can be used to identifyB[t] as a subobject of B�∞ . Under this identification, the diagram b : �op

+ → B[t] ↪→ B�∞

is the compositeb := �∞ × �op

+⊕−→ �∞

b−→ B.

The vertex [−1] ∈ �op+ acts as the identity for the join operation, so the algebra b[−1]

equals b ∈ B[t]. The vertex [0] ∈ �op+ acts by precomposition with − ⊕ [0] : �∞ → �∞,

restricting b to the subdiagram

tb ηtb //t2bµboo

ηttb //

tηtb // t3bµtboo

tµboo

ηtttb //

tηttb //

ttηtb //t4b · · · · · ·

µttboo

tµtboo

ttµboo

∈ B

that defines the free algebra tb ∈ B[t]. The other vertices of �op+ act by further restriction.

In summary, the functor b : �op+ → B[t] carries the vertex [n] ∈ �op

+ to tn+1b ∈ B[t]. Asimilar analysis can be used to identify the morphisms in b as components of the original


diagram b : �∞ → B. In conclusion, we write

b tbβoo tηb // t2btβoo

µboo tηtb //

ttηb //t3b · · · · · ·tµboo

ttβoo

µtboo

∈ B[t]

for the obvious 1-skeletal subset of t•b = b ∈ B[t]�op+ .

6.3.17. Theorem (canonical colimit representation of algebras). Given a homotopy co-herent monad T : Mnd → qCat∞, the functor t• : B[t] → B[t]�

op+ of (6.3.16) encodes an

absolute left lifting diagram

B[t]

c��

B[t]

id;;

t•// B[t]�

op⇑ =

B[t]

c��

B[t]t•// B[t]�

op+

ev−1

::

res// B[t]�

op⇑ (6.3.18)

created from the ut-split simplicial objects in B

⇑

B[t]

c��

ut // B

c��

B[t]t•//

id;;

B[t]�op

(ut)�op// B�op

=⇑

B

c��

B[t] s•// B�∞

ev0

;;

res// B�op

(6.3.19)

The colimits (6.3.18) exhibit the algebras for a homotopy coherent monad T on the quasi-category B as colimits of canonical simplicial objects whose vertices are free algebras.

Proof. An immediate corollary of proposition 6.3.7: simply pre-compose the absolute lift-ing diagrams (6.3.11) and (6.3.12) by the functor B[t] → S(ut) constructed in observa-tion 6.3.15. �

7. Monadicity

Our aim in this section is to provide a new proof of the quasi-categorical monadicitytheorem, originally due to Lurie [19]. Given an adjunction of quasi-categories, theorem4.3.11 extends this data to a homotopy coherent adjunction from which we can constructan associated homotopy coherent monadic adjunction, as described in definition 6.1.14.Immediately from our weighted limits definition, there is a comparison map from theoriginal adjunction to the monadic one, defined as a component of a simplicial naturaltransformation between the Adj-indexed simplicial functors but of course also interpretablein qCat2. The monadicity theorem provides conditions on the original adjunction underwhich this comparison functor is an equivalence quasi-categories.

72 RIEHL AND VERITY

7.1. Comparison with the monadic adjunction. Suppose given a homotopy coherentadjunction H : Adj → qCat∞ which restricts to a homotopy coherent monad T : Mnd →qCat∞. By the Yoneda lemma, the limits of the diagramH weighted by the two representa-bles Adj+ and Adj− are isomorphic to the two objects in the diagram. Furthermore, theYoneda embedding Adj∗ : Adj

op → sSetAdj defines an adjunction between these weights,whose left adjoint is a map Adj− → Adj+. Applying {−, H}Adj returns the homotopycoherent adjunction:

A ∼= {Adj−, H}Adju

22⊥ {Adj+, H}Adj∼= B

frr

7.1.1. Observation (changing the index for the weights). In order to compare the monadicadjunction defined in 6.1.14 with H it will be convenient to have formulas that describe themonadic adjunction as a weighted limit indexed over the simplicial category Adj insteadof its subcategory Mnd. As a consequence of lemma 5.1.11, this can be done by simplytaking the left Kan extension of the defining weights.

Note that if W is projectively cofibrant, so is lanW : the right adjoint in the adjunction

sSetAdj

res

22⊥ sSetMndlan

rr

manifestly preserves trivial fibrations, so the left adjoint preserves cofibrant objects.

7.1.2. Definition (quasi-category of algebras, revisited). Recall from definition 6.1.7 thatthe quasi-category of algebras is the limit of the homotopy coherent monad underlying Hweighted by W−, the restriction of the representable Adj−. By lemma 5.1.11, this quasi-category is equivalently described as the limit of the H weighted by lanW−

AdjlanW−

""⇑∼=

MndW−

//-

<<

sSeti.e., {W−, resH}Mnd

∼= {lanW−, H}Adj.

The left Kan extension of the representable functor W+ on Mnd along the inclusionMnd → Adj is the representable Adj+. Because the inclusion Mnd ↪→ Adj is full,lanW−(+) ∼= W−(+) = �∞. By the standard formula for left Kan extensions, the value oflanW− : Adj→ sSet at the object − is defined by the coend

lanW−(−) =

∫ Mnd

Adj(+,−)× �∞ = coeq(�−∞ × �+ × �∞

//// �−∞ × �∞

)This coequaliser identifies the left and right actions of �+. In accordance with our nota-tional conventions, the categories on the right denote the quasi-categories given by theirnerves, but because h : qCat → Cat preserves finite products as well as colimits, our re-marks apply equally to the analogous coequaliser in Cat.


7.1.3. Lemma. The coequaliser of the pair

�−∞ × �+ × �∞//// �−∞ × �∞ (7.1.4)

identifying the left and right actions of �+ is �op.

Proof. The coequaliser (7.1.4) is computed pointwise in Set. We will show that the obviousdiagram �−∞×�+×�∞ ⇒ �−∞×�∞ → �op is a coequaliser diagram using the graphicalcalculus for the simplicial category Adj.

To see that the map from the coequaliser to �op is surjective on k-simplices note thata k-simplex in �op

+ = Adj(−,−) lies in the subcategory �op if and only if its representingsquiggle passes through the level labelled +. The squiggle to the left of this point is ak-simplex in �−∞ and the squiggle to the right is a k-simplex in �∞. Juxtaposition definesa surjective map.

To see the map from the coequaliser is injective, suppose a squiggle representing a k-simplex in �op passes through + at least once and consider the two preimages in �−∞×�∞corresponding to distinct subdivisions. The squiggle between the two chosen + symbols isa k-simplex in �+. Hence, the coequaliser already identifies our two chosen preimages ofthe given k-simplex of �op. �

7.1.5. Observation. To summarise the preceeding discussion, we can give the followingexplicit description of the weight lanW− as a simplicial functor Adj → qCat∞. Therepresentable Adj− : Adj→ qCat∞ defines an adjunction

Adj(−,−)u◦−

22⊥ Adj(−,+)

f◦−rr ! �op

+u◦−

22⊥ �∞

f◦−rr

We see that the left adjoint lands in the full subcategory �op ⊂ �op+ by employing our

graphical calculus: �op ⊂ �op+ is the simplicial subset consisting of squiggles from − to

− that go through +. The functor f ◦ − post-composes a squiggle from − to + withf = (−,+), and hence lands in this subcategory. The restricted adjunction is lanW−.

lanW− : Adj→ qCat∞ ! �op

u◦−22⊥ �∞

f◦−rr (7.1.6)

7.1.7. Definition (monadic adjunction, revisited). Enriched left Kan extension defines asimplicial functor lan : sSetMnd → sSetAdj. Composing with (6.1.15) yields a simplicialfunctor Adjop → sSetAdj defining an adjunction between the weights lanW− and Adj+.Composing with {−, H}Adj produces the monadic adjunction

B[t] ∼= {lanW−, H}Adjut

22⊥ {Adj+, H}Adj∼= B

f t

rr

74 RIEHL AND VERITY

7.1.8. Remark. The weight (7.1.6) for the monadic adjunction is the same adjunction usedto prove theorem I.5.3.1. Its data demonstrates that any augmented simplicial objectadmitting a splitting defines a colimit cone over its underlying simplicial object.

7.1.9. Observation (comparison with the monadic adjunction). The counit of the left Kanextension–restriction adjunction induces a comparison from the weight for the monadic ho-motopy coherent adjunction to the Yoneda embedding. The components of this simplicialnatural transformation are maps

lanW− = lan resAdj− → Adj− and Adj+ ∼= lan resAdj+ → Adj+the latter of which we take to be the identity. Taking limits with these weights producesa commutative diagram

A ∼= {Adj−, H}Adj //

u

((

{lanW−, H}Adj∼= B[t]

utuuB ∼= {Adj+, H}Adj

f

hh f t55

(7.1.10)

in which the comparison functor A → B[t] commutes with both the left and the rightadjoints.

A second justification follows: the monadic homotopy coherent adjunction is isomorphicto the right Kan extension of the restriction of H to Mnd; this can be seen for instancefrom the weight limits formula for (simplicial) right Kan extensions. The comparison mapis induced by the universal property of the right Kan extension.

7.2. The monadicity theorem. Write R : A→ B[t] for the comparison functor definedby (7.1.10). The monadicity theorem supplies conditions under which R is an equivalenceof quasi-categories. The first step is to show that if the quasi-category A admits certaincolimits, then R has a left adjoint.

7.2.1. Definition (u-split simplicial objects). As in (6.3.14), if we apply the contravariantfunctor {−, H}Adj to the left-hand pushout

Adj+ × �op� _

Adj+×(u◦−)

��

� � (u◦−)×�op// Adj− × �op

� _

��

S(u)

��

// B�∞

Bu◦−

��

Adj+ × �∞ // Ws A�op

u�op// B�op

(7.2.2)

in sSetAdj we obtain the right hand pullback in qCat∞. In other words, if we take thelimit of the homotopy coherent adjunction H weighted by this weight Ws we obtain thequasi-category S(u) of u-split simplicial objects associated with the right adjoint of thatadjunction. In contrast with definition 6.3.5, a vertex in S(u) is a simplicial object whoseimage under u admits both an augmentation and a splitting.


We use the same notation Ws for the weights defined by (6.3.14) and (7.2.2); contextwill disambiguate these. The left-hand vertical of the pullback defining this quasi-categoryS(u) is a family of diagrams k : S(u)→ A�op

.Using definition I.5.2.9, we say the quasi-category A admits colimits of u-split simplicial

objects if there exists a functor colim: S(u) → A and a 2-cell that define an absolute leftlifting diagram

A

c��

S(u)k//

colim

;;

A�op⇑

(7.2.3)

7.2.4. Theorem (monadicity I). Let H be a homotopy coherent adjunction with underlyinghomotopy coherent monad T and suppose that A admits colimits of u-split simplicial objects.Then the comparison functor R : A→ B[t] admits a left adjoint.

Proof. We begin by defining the left adjoint L : B[t] → A. The weight lanW− defines acone under (7.2.2):

Adj+ × �op

��

// Adj+ × �∞

��

Adj− × �op`// lanW−

(7.2.5)

By the Yoneda lemma, the map Adj+ × �∞ → lanW− is determined by a map �∞ →lanW−(+) = �∞, which we take to be the identity. Similarly, the map Adj−×�op → lanW−is determined by a map �op → lanW−(−) = �op which we also take to be the identity.The cone (7.2.5) defines a map of weights Ws → lanW− and hence a map B[t]→ S(u) ofweighted limits. Define L to be the composite of this functor with colim: S(u) → A. Itfollows from commutativity of (7.2.5) that L : B[t] → A commutes with the left adjointsf t and f .

The diagram (7.2.3) defining the colimits of u-split simplicial objects in A restricts todefine an absolute left lifting diagram

A

c��

B[t] //

L

<<

A�op⇑

(7.2.6)

The diagram component B[t] → A�opis derived from the map ` of weights. Recall the

functor R : A → B[t] is derived from the counit of the left Kan extension–restriction

76 RIEHL AND VERITY

adjunction, a map ν : lanW− → Adj− of weights. The left-hand diagram of weights

Adj− Adj− × �op+

idAdj− ×!oo

Ac //

R

��

A�op+

res

��

lanW−

ν

OO

Adj− × �op

OO

`oo B[t] // A�op

commutes because the lower-left composite corresponds, via the Yoneda lemma, to theinclusion �op ↪→ �op

+ , as does the upper-right composite. Hence, the induced functorsdefine a commutative diagram on weighted limits, displayed above right.

In this way we see that the canonical natural transformation

A

c��

A�op+

ev−1

;;

res// A�op⇑

defined by the 2-cell (6.3.4) induces the 2-cell on the right-hand side that we call ε : LR⇒ idvia the universal property of the absolute left lifting diagram (7.2.6):

A

c��

id //

=

A

c��

A�op+

ev−1

;;

res// A�op⇑

=:

A

R��

id //

⇑

A

c��

B[t] // A�op

=

A

R��

id //

∃!⇑εA

c��

B[t]L

<<

// A�op⇑

Similarly, the commutative diagram of weights induces a commutative diagram of limits

lanW−

B[t]

��

t•

$$

Adj− × �op

OO

lanW− × �op

hh

ν×id�opoo A�op

R�op// B[t]�

op

Hence, the composite 2-cell

A

c��

R // B[t]

c��

B[t] //

L

<<

A�op⇑

R// B[t]�

op

=:

B[t]RL //

id��

⇑

B[t]

c��

B[t] // B[t]�op

=

B[t]RL //

id��

∃!⇑η

B[t]

c��

B[t]

id

;;

// B[t]�op

⇑

defines the 2-cell η : id⇒ RL using the universal property of the canonical colimit diagramof theorem 6.3.17.

It follows from a straightforward appeal to the uniqueness statements of these univer-sal properties, left to the reader, that the 2-cells defined in this way satisfy the triangleidentities and hence give rise to an adjunction L a R in qCat2. �


7.2.7. Theorem (monadicity II). If A admits and u : A→ B preserves colimits of u-splitsimplicial objects, then the unit of the adjunction L a R of theorem 7.2.4 is an isomorphism.If u is conservative, then

AR

22⊥ B[t]L

rr

is an adjoint equivalence.

Proof. Recall u factors as ut ·R. The first hypothesis asserts that

A

c��

R // B[t]ut //

c��

B

c��

B[t] //

L

<<

A�op⇑

R�op// B[t]�

op

(ut)�op// B�op

is an absolute left lifting diagram. By inspecting the defining maps of weights, we see thatthe bottom composite B[t]→ B[t]�

opis the map t• of (6.3.16). By theorem 6.3.17,

B[t]

c��

ut // B

c��

B[t]t•//

idB[t];;

B[t]�op

⇑

(ut)�op// B�op

is also an absolute left lifting diagram.This gives us two a priori distinct absolute left lifting diagrams defining colimits for the

same family of diagrams B[t]→ B�op. Write α for the first 2-cell and β for the second. By

the definition of η in the proof of theorem 7.2.4, we have

B[t]ut //

⇑α

B

c��

B[t]

RL==

// B�op

=

B[t]

idB[t]

��

RL //

⇑ηB[t]

⇑β

ut // B

c��

B[t]

idB[t]

==

// B�op

Conversely, the universal property of α can be used to define a 2-cell ζ

B

c��

B[t]

ut<<

// B�op⇑β

=

B[t]⇑ζ

⇑α

ut //

idB[t]

��

B

c��

B[t] //

utRL

<<

B�op

It is easy to see that ζ and utη are inverses. But corollary 6.2.3 tells us that ut : B[t]→ Bis conservative. Because isomorphisms in quasi-categories are defined pointwise, it followsthat (ut)B[t] : B[t]B[t] → BB[t] is as well. Thus η is an isomorphism.

Isomorphisms are preserved by restricting along any functor. In particular, ηR is anisomorphism. By uniqueness of inverses, Rε : RLR ⇒ R is its inverse, and is also an

78 RIEHL AND VERITY

isomorphism. Hence uε = utRε is an isomorphism. If u is conservative, it follows that ε isan isomorphism, and hence that L a R is an adjoint equivalence. �

References

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Department of Mathematics, Johns Hopkins University, Baltimore, MD 21218, USAE-mail address: [email protected]

Centre of Australian Category Theory, Macquarie University, NSW 2109, AustraliaE-mail address: [email protected]

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